Automatic Computing Methods for Special Functions. Part IV. Complex Error Function, Fresnel Integrals, and Other Related Functions

Irene A Stegun; Ruth Zucker

doi:10.6028/jres.086.031

. 1981 Nov-Dec;86(6):661–686. doi: 10.6028/jres.086.031

Automatic Computing Methods for Special Functions. Part IV. Complex Error Function, Fresnel Integrals, and Other Related Functions

Irene A Stegun ^1,^*, Ruth Zucker ^1,^*

PMCID: PMC6753013 PMID: 34566066

Abstract

Accurate, efficient, automatic methods for computing the complex error function to any precision are detailed and implemented in an American Standard FORTRAN subroutine. A six significant figure table of erfc z, $e^{x^{2}}$ erfc z, and $e^{z^{2}}$ erfc(−z) is included for z in polar coordinate form with the modulus of z ranging from 0 to 9. The argand diagram is given for erf z.

Keywords: Argand diagram, complex error function, continued fraction, Dawson’s function, FORTRAN subroutine, Fresnel integrals, key values, line broadening function, plasma dispersion function, Voigt function

1. Introduction

In computing many of the functions of mathematical physics, for example, Fresnel integrals, Dawson’s integral, Voigt function, plasma dispersion function, etc., difficulties are frequently encountered. Since these functions may be expressed in terms of the error function of complex argument, we have chosen this function for Part IV.¹ The major part of the coding of the power series, continued fraction and asymptotic expansion computations for complex arguments will carry over equally well for other functions.

As Part I was devoted to the error function of a real variable, the probability function and other related functions, Part IV will only emphasize those functions and pitfalls due to complex arguments.

While accuracy over the entire domain of definition remains our main concern, the methods employed ensure efficiency, portability and ease of programming and modification.

If one supplies approximate values for the maximum machine value, minimum machine value, the upper bound of the sine, cosine routine, and the upper bound to the acceptable relative error and gives the square root of π to the required number of significant figures, the detailed methods will work for computations ranging from very low precision to multi-precision.

The argand diagram of erf z is included as well as the implementing ANS FORTRAN program and a six significant figure table of erfc z, $e^{x^{2}}$ erfc z and $e^{z^{2}}$ erfc( − z) for z in polar coordinate form with the modulus of z ranging from 0 to 9.

2. Mathematical properties

Relevant formulas are collected here for completeness and ease of reference. In keeping with the convention of the Handbook [1],² z = x + i y is a complex variable.

A. Definitions

\erf z = \frac{2}{\sqrt π} \int_{0}^{z} e^{- t^{2}} d t erfc z = \frac{2}{\sqrt π} \int_{z}^{\infty} e^{- t^{2}} d t = 1 - \erf z

(The path of integration is subject to the restriction arg t → α with $| α | < \frac{π}{4}$ as t →∞ along the path. If Rt² remains bounded to the left, $α = \frac{π}{4}$ is permissible.)

w (z) = e^{- z^{2}} (1 + \frac{2 i}{\sqrt π} \int_{0}^{z} e^{t^{2}} d t) = e^{- z^{2}} erfc (- i z) = e^{ξ^{2}} erfc ξ (ξ = - i z) = \frac{i}{π} \int_{- \infty}^{\infty} \frac{e^{- t^{2}} d t}{z - t} = \frac{2 i z}{π} \int_{0}^{\infty} \frac{e^{- t^{2}} d t}{z^{2} - t^{2}} (I z > 0)

with $F (z) = e^{- z^{2}} \int_{0}^{z} e^{t^{2}} d t$ (Dawson’s Function)

and $Z (z) = \frac{1}{\sqrt π} \int_{- \infty}^{\infty} \frac{e^{- t^{2}} d t}{t - z}$ (Plasma Dispersion Function)

E (z) = C (z) + i S (z) = \int_{0}^{z} e^{i x t^{2} / 2} d t = \int_{0}^{z} \cos (\frac{π t^{2}}{2}) d t + i \int_{0}^{z} \sin (\frac{π t^{2}}{2}) d t = \frac{1 + i}{2} \erf [\frac{\sqrt π}{2} (1 - i) z] (Fresnel Functions)

W (x, t) = \frac{ω}{{(4 π t)}^{1 / 2}} \int_{- \infty}^{\infty} \frac{e^{- u^{2}} d u}{u^{2} + ω^{2}} = {(\frac{π}{4 t})}^{1 / 2} e^{ω^{2}} erfc ω

where

ω = (1 - i x) / 2 t^{1 / 2} = U (x, t) + i V (x, t) (Voigt Function) = \frac{1}{{(4 π t)}^{1 / 2}} \int_{- \infty}^{\infty} \frac{e^{- {(x - y)}^{2} / 4 t}}{1 + y^{2}} d y + \frac{i}{{(4 π t)}^{1 / 2}} \int_{- \infty}^{\infty} \frac{y e^{- {(x - y)}^{2} / 4 t}}{1 + y^{2}} d y H (a, u) = \frac{a}{π} \int_{- \infty}^{\infty} \frac{e^{- t^{2}} d t}{{(u - t)}^{2} + a^{2}} = \frac{1}{a \sqrt π} U (u / a, 1 / 4 a^{2}) (Line Broadening Function)

B. Series Expansions

\erf z = \frac{2}{\sqrt π} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} z^{2 n + 1}}{n! (2 n + 1)} = \frac{2}{\sqrt π} e^{- z^{2}} \sum_{n = 0}^{\infty} \frac{2^{n} z^{2 n + 1}}{1 \cdot 3... (2 n + 1)}

C. Continued Fraction (R_z> 0)

e^{z^{2}} erfc z = \frac{1}{\sqrt π} [\frac{1}{z +} \frac{1 / 2}{z +} \frac{1}{z +} \frac{3 / 2}{z +} \frac{2}{z +} \dots] = \frac{2 z}{\sqrt π} [\frac{1}{2 z^{2} + 1} - \frac{1 \cdot 2}{2 z^{2} + 5} - \frac{3 \cdot 4}{2 z^{2} + 9} - \dots] (“ Even ” Form)

D. Asymptotic Expansion

e^{z^{2}} erfc z \sim \frac{1}{\sqrt π} [\frac{1}{z} + \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n} 1 \cdot 3 \dots (2 n - 1)}{z {(2 z^{2})}^{n}}] (z \to \infty, | \arg z | < \frac{3 π}{4})

E. Symmetry Relations

\erf (- z) = - \erf z \erf \bar{z} = \bar{\erf z} w (- z) = 2 e^{- z^{2}} - w (z) w (\bar{z}) = \bar{w (- z)} C (- z) = - C (z), S (- z) = - S (z) C (i z) = i C (z), S (i z) = - i S (z) C (\bar{z}) = \bar{C (z)}, S \bar{(z)} = \bar{S (z)}

3. Method

The main functions under consideration are the error function ERFZ, the complementary error function ERFCZ, and the exponential of z² times the complementary error function EZ2CZ. All other functions may be obtained from these three. To simplify testing, computations are performed for z in the first quadrant AZ and symmetry relations are then employed to make adjustments for other quadrants. For the special case z = 0, no computations are performed and the following function values are returned: ERFZ = 0, ERFCZ = 1 and EZ2CZ = 1.

Real type variables are used throughout to readily allow for double precision computation if greater accuracy is needed. The machine dependent constants are placed in a labeled section at the beginning of the subroutine. Function references are likewise grouped together when possible and attention called to the statement labels of the remaining function references. Real and imaginary parts of complex variables have R and I as final characters.

Since EZ2CZ for z in the first quadrant is machine representable even with the real and imaginary parts of z equal to the maximum machine value CMAX (provided its reciprocal is larger than the minimum machine value CMIN), checking for the range of the argument z has been omitted. However, the extensive range necessitates a fair amount of testing for overflows. Underflows are assumed to be set to zero. Overflows are set equal to the maximum machine value and an error indicator IERR set for the number of functions affected. If only EZ2CZ lies outside the machine range, IERR = 1, otherwise IERR = 3. As often as possible, computations are arranged so as to give the correct results for the three functions if they lie within the range of the machine.

In computing the modulus RHO = √(AZR)² + (AZI)² of a complex quantity AZ = AZR + i AZI in the first quadrant, RHO may lie in the machine range but (AZR)², (AZI)² or their sum may be outside the range. We select the larger ARIMX and smaller ARIMN of either AZR or AZI, and compute the ratio RMNMX = ARIMN/ARIMX. A factor of RHO called PRHO is computed as the square root of (RMNMX*RMNMX + ONE). This factor, which is greater than or equal to one and less than or equal to the square root of 2, can then be used to check for overflow. The quantity ARIMX must be less than CMAX divided by PRHO for RHO to lie in the machine range. A similar procedure is followed in computing the real and imaginary parts of (AZ)² = Z2R + i AZ2I = (AZR)² − (AZI)² + i 2AZR*AZI with first checking to ensure ARIMX is greater than or equal to 1.

Analysis has indicated and testing confirmed that the power series PS is most useful from the standpoint of accuracy and efficiency for RHO less than RHOLS( = 1.5) and when AZR is less than or equal to 1 provided RHO is less than AELL (=√ −1 n(TOLER)) where TOLER is the upper limit for the relative error. The continued fraction expansion CF is most useful for AZR greater than 1 and RHO greater than or equal to RHOLS. The asymptotic expansion AE is most useful for AZR less than or equal to 1 for RHO greater than or equal to AELL. For RHO greater than or equal to RHOLC (=√0.5/TOLER) a rearrangement of arithmetic operations for the first term of the asymptotic expansion is necessary to maintain the accuracy of EZ2CZ. In the continued fraction and asymptotic expansion regions only, EZ2CZ is first computed; it tends to zero for large |z| and the exponential of −z² tends to infinity for small AZR. To maintain accuracy here, we compute the exponential of − Z2R/6 and do continuous multiplication and testing with appropriate factors to obtain ERFC. The imaginary part of (AZ)² is tested against ULSC, the upper limit of the sine, cosine routine.

Figure 1 below maps the regions for the various methods.

The dividing line RHO = AELL between the use of the power series and the asymptotic expansion and RHO = RHOLC are the only boundaries subject to the required precision. Single and double precision results, for example on the Univac 1108, are the results of two different methods in the region where RHO roughly lies between 4.3 and 6.4.

This mapping of the region ensures for the required precision that the least number of terms are computed and the loss of significance is kept to a minimum. While the second form of the power series is preferable for real positive z, since all terms are positive, the first form ensures greater accuracy for complex z since the real and imaginary parts of the terms may be positive, negative or zero for any RN( = n). The power series is there more rapidly convergent; the relative error may then be approximated by the ratio of the term to the sum of terms; comparison of this approximant with the tolerance for the relative error controls the number of terms needed. Since the terms tend to zero through underflow, there will always be convergence even if the tolerance is made considerably smaller than the precision of the machine. The power series is evaluated using recurrence relations in the following form:

ERFZ = \frac{TWO}{SQRT (PI)} \sum_{RN = 0}^{RNF} SGN (RN) * TM (RN) = \frac{TWO}{RTPI} * SUM

where

SGN (0) = 1, SGN (RN + 1) = - SGN (RN) TM (RN) = ((AZ * * (2 * RN + 1)) / 1 * 2 * \dots RN) (2 * RN + 1) TM (RN) = PTM (RN) / DN (RN) PTM (0) = AZ, PTM (RN + 1) = (AZ * * 2) * PTM (RN) (RN + 1) DN (0) = 1, DN (RN + 1) = DN (RN) + 2

In determining the terminal value of RN, normalization is necessary to avoid overflows and underflows. The normalization factor TMAX is the maximum of the absolute value of the real and imaginary parts of TM and SUM. If TMAX equals zero or |TM/TMAX|² underflows, RNF = RN.

The continued fraction expansion starts to converge more slowly as z tends to zero. The “even” form is used since the required number of terms is halved at the expense of very little extra computation for successive numerators and denominators. The continued fraction is evaluated by using the recurrence relations in the “forward” direction. The number of terms needed is determined by checking to see if the relative error of two successive convergents is less than the tolerance. On the other hand, if the relative error remains constant or starts to increase, the recurrence is terminated and the prior convergent taken as the value of the continued fraction. In this way, the process is always terminated when maximum precision is attained.

The “even” form of the continued fraction takes on the following implementation:

EZ 2 CZ = \frac{2 * AZ}{RTPI} \prod_{RN = 1}^{RNF} \frac{AM (RN)}{BM (RN)}

with AM (1) = 1, AM (RN + 1) = - WM (RN + 1) * (WM (RN + 1) + 1) BM (1) = 2 * (AZ * * 2) + 1, BM (RN + 1) = BM (RN) + 4

where WM (1) = - 1, WM (RN + 1) = WM (RN) + 2 EZ 2 CZ = (AZ * (FM / GM)) * 2 / RTPI = (AZ * F (RN)) * 2 / RTPI

where FM (- 1) = 1, FM (0) = 0 GM (- 1) = 0, GM (0) = 1

and FM (RN) = BM (RN * FM (RN - 1) + AM (RN) * FM (RN - 2) GM (RN) = BM (RN) * GM (RN - 1) + AM (RN) * GM (RN - 2)

The relative error may be approximated by [F(RN) – F(RN – 1)]/F(RN) = RE(RN). If the modulus squared of the relative error REM2(RN) is less than the square of the tolerance divided by 8, RNF = RN. If REM2(RN) is greater than or equal to REM2(RN – 1), then RNF = RN – 1. Normalization is likewise necessary here to avoid overflows in computing the relative error and its modulus squared and also in the generation of the successive convergents.

The asymptotic expansion is likewise evaluated using recurrence relations in the following form:

EZ 2 CZ = \frac{1}{SQRT (PI)} \sum_{RN = 0}^{RNF} SGN (RN) * TM (RN) = SUM / RTPI

where SGN (0) = 1, SGN (RN + 1) = - SGN (RN) TM (RN) = (1 / AZ) * (1 * 3 * \dots (2 * RN - 1)) / (2 * (AZ * * 2)) * * RN TM (0) = 1 / AZ, TM (RN + 1) = DN (RN + 1) * TM (RN) * (1 / (2 * (AZ * * 2)))

with DN (1) = 1, DN (RN + 1) = DN (RN) + 2.

The relative error may be approximated here by the ratio TM/SUM. The convergence test precedes the divergence test and is implemented as REM2 less than (TOLER**2)/8 to attain greater accuracy in both the real and imaginary parts. If the modulus squared of the term remains the same or increases, the prior sum is taken as the final sum.

For z along the imaginary axis, the error function is purely imaginary; the real part of erfc z = 1 and of $e^{z^{2}}$ $erfc z = e^{- {(AZI)}^{2}}$ . No difficulties arise in the use of the power series. However, since the asymptotic expansion is given for |z| → ∞, the correction must be applied for AZR → 0.

The following table gives an indication of the number of terms needed to obtain maximum machine accuracy on the Univac 1108 with the various methods of computation.

Method	Number of Terms

	Single Precision	Double Precision
	TOLER = .745E-8	TOLER = .867D-18

Power Series	50	112
Continued Fraction	25	99
Asymptotic Expansion	22	45

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4. Range

If the real part of z is zero or positive, $e^{z^{2}}$ erfc z is valid for z throughout the entire machine range. Otherwise, the real part of z² is essentially limited by the range of the exponential library subroutine with the imaginary part of z² limited by the range of the sine, cosine library subroutine.

5. Accuracy and Precision

The maximum relative error, generally in erfc z, except for regions in the immediate neighborhood of zeros of the real and/or imaginary parts of the functions is 8E-6 for single precision computation on the Univac 1108.

The precision may be varied by changing the value of TOLER.

6. Timing (Univac 1108 Time/Sharing Executive System)

The time estimates given below are highly dependent on the operating system environment and consequently should not be relied on for critical timing measurements.

	Time (Seconds)
	Single Precision	Double Precision
Region	TOLER = .745E-8	TOLER = .867D-18
ZR = 0(.1)4, ZI = 0(.2)8 (1681 values)	5.94	22.5
Method	Maximum Time/Evaluation

Power Series	.0101	.038
Continued Fraction	.0088	.052
Asymptotic Expansion	.0035	.0093

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7. Testing

The language of the subroutine was checked for conformity with the PFORT VERIFIER.³ Test arguments were devised and used in the analysis of the subroutine with the PROFILER.⁴

The subroutine was used to obtain related functions which were checked against available published tables ([1]-[5], [7]-[9], [13], [15]). Single precision results covering the 9×9 grid were compared against double precision results. This precision test particularly verified that the scaling operations were valid and undetected overflows had not occurred.

Where applicable, results were obtained by independent methods, for example, the power series and continued fraction, the power series and asymptotic expansion or all three methods. Otherwise, both forms of the power series or continued fraction were used. In addition, numerical integration of various forms of the integral representation was employed.

8. Test Values

Six-significant figure tables of erfc z, $e^{z^{2}}$ erfc z and $e^{z^{2}}$ erfc(−z) are included in the appendix. The computations were performed with double precision arithmetic to provide more accurate values for checking purposes. The arguments are in polar coordinate form $z = ϱ e^{i θ}$ for ϱ = 0(.02).2(.1)3(.5)9, θ = 0°(15°)30°, 37.5 °, 45°, 50 °(10°)90°. Values of the functions for z in other quadrants are readily obtainable with symmetry relations.

9. Argand Diagram of erf $z = \frac{2}{\sqrt π} \int_{0}^{z = ρ e^{i θ}} e^{- t^{2}} d t$

Since erf z and erfc z are complementary functions, erf z is not tabulated but the argand diagram of the function is included to illustrate divergent and convergent spirals.

10. Special Values

Relevant values are collected here for completeness and ease of modification and checking of the program.

Zeros

	erf z_s = 0
s	z_s
1	1.45061616 + i 1.88094300
2	2.24465927 + i 2.61657514
3	2.83974105 + i 3.17562810

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	w(z_s) = 0
s	z_s
1	1.99146684 − i 1.35481013
2	2.69114902 − i 2.17704491
3	3.23533087 − i 2.78438761

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	C(z_s) = 0	S(z_s) = 0
s	z_s	z_s
1	1.7437 + i .3057	2.0093 + i .2885
2	2.6515 + i .2529	2.8335 + i .2443
3	3.3204 + i .2240	3.4675 + i .2185

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Maxima and Minima of Fresnel Integrals

s	$M_{s} = C (\sqrt{4 s + 1})$	$m_{s} = C (\sqrt{4 s + 3})$	$M_{s}^{*} = S (\sqrt{4 s + 2})$	$m_{s}^{*} = S (\sqrt{4 s + 4})$
0	.779893	.321056	.713972	.343416
1	.640807	.380391	.628940	.387969
2	.605721	.404261	.600362	.408301
3	.588128	.417922	.584942	.420516

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Radius of Univalence ϱ

\int_{0}^{z} e^{- t^{2}} d t e^{z^{2}} \int_{0}^{z} e^{- t^{2}} d t ϱ = 1.5748376 .92413887

Maximum and Inflection Point for Dawson’s Integral

F(.92413 88730) = .54104 42246 7

F(1.50197 52682) = .42768 66160

Related Constants

\begin{array}{r} 1^{°} = 1.74532 92519 94329 57692 36907 68488 61271 (- 2) r \\ π = 3.14159 26535 89793 23846 26433 83279 50288 \\ π / 2 = 1.57079 63267 94896 61923 13216 91639 75144 \\ \sqrt π = 1.77245 38509 05516 02729 81674 83341 14518 \\ \sqrt π / 2 = .88622 69254 52758 01364 90837 41670 57259 \\ 2 π = 6.28318 53071 79586 47692 52867 66559 00576 \\ 2 \sqrt π = 3.54490 77018 11032 05459 63349 66682 29036 \\ e = 2.71828 18284 59045 23536 02874 71352 66249 \\ 1 / π = .31830 98861 83790 67153 77675 26745 02872 \\ 2 / π = .63661 97723 67581 34307 55350 53490 05744 \\ 1 / \sqrt π = .56418 95835 47756 28694 80794 51560 77258 \\ 2 / \sqrt π = 1.12837 91670 95512 57389 61589 03121 54517 \\ 1 / 2 π = .15915 49430 91895 33576 88837 63372 51436 \\ 1 / 2 \sqrt π = .28209 47917 73878 14347 40397 25780 38629 \\ 1 / e = .36787 94411 71442 32159 55237 70161 46086 \\ erf 1 = .84270 07929 49714 86934 12206 35082 60926 \\ \int_{0}^{1} e^{- t^{2}} d t = (\sqrt π / 2) \erf 1 = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{n! (2 n + 1)} = .74682 41328 12427 02539 94674 36131 85300 \\ (\sqrt π / 2) e^{1} \erf 1 = \sum_{n = 0}^{\infty} \frac{2^{n}}{1 \cdot 3... (2 n + 1)} = 2.03007 84692 78704 97553 90899 25665 95044 \\ \int_{0}^{1} e^{t^{2}} d t = \sum_{n = 0}^{\infty} \frac{1}{n! (2 n + 1)} = 1.46265 17459 07181 60880 40485 86856 98815 \\ \sum_{n = 0}^{\infty} \frac{1}{(2 n)! (4 n + 1)} = 1.10473 79393 59804 31710 17580 11494 42058 \\ \sum_{n = 0}^{\infty} \frac{1}{(2 n + 1)! (4 n + 3)} = .35791 38065 47377 29170 22905 75362 56757 \\ \sum_{n = 0}^{\infty} \frac{2^{2 n}}{1 \cdot 3... (4 n + 1)} = 1.28407 89880 95736 69733 77386 73036 75360 \\ \sum_{n = 0}^{\infty} \frac{2^{2 n + 1}}{1 \cdot 3... (4 n + 3)} = .74599 94811 82968 27820 13512 52629 19684 \\ e^{- 1} \int_{0}^{1} e^{t^{2}} d t = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} 2^{n}}{1 \cdot 3... (2 n + 1)} = .53807 95069 12768 41913 63874 20407 55675 \end{array}

Typical Tolerances and Their Natural Logarithms

2⁻¹⁶ =	0.15258 78906 25(−4)
2⁻²⁴ =	.59604 64477 53906 25(−7)
2⁻²⁷ =	.74505 80596 92382 8125(−8)
2⁻³⁶ =	.14551 91522 83668 51806 64062 5(−10)
2⁻⁴⁸ =	.35527 13678 80050 09293 55621 33789 0625(−14)
2⁻⁵⁶ =	.13877 78780 78144 56755 29539 58511 35253 90625(−16)
2⁻⁶⁰ =	.86736 17379 88403 54720 59622 40695 95336 91406 25(−18)
2⁻¹⁰⁸ =	.30814 87911 01957 73648 89564 70813 58837 09660 96263 71446 21112 38390 20729 06494 14062 5(−32)

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log_e(2⁻¹⁶) =	−11.09035 48889 59124 95067 57139 43330 82508
log_e(2⁻²⁴) =	−16.63553 23334 38687 42601 35709 14996 23763
log_e(2⁻²⁷) =	−18.71497 38751 18523 35426 52672 79370 76733
log_e(2⁻³⁶) =	−24.95329 85001 58031 13902 03563 72494 35645
log_e(2⁻⁴⁸) =	−33.27106 46668 77374 85202 71418 29992 47526
log_e(2⁻⁵⁶) =	−38.81624 21113 56937 32736 49988 01657 88781
log_e(2⁻⁶⁰) =	−41.58883 08335 96718 56503 39272 87490 59408
log_e(2⁻¹⁰⁸) =	−74.85989 55004 74093 41706 10691 17483 06935

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Maximum and Minimum Machine Values and Their Natural Logarithms (Univac 1108 Single and Double Precision Limits)

NBC = Number of binary digits in the (biased) characteristic of a floating point number

2^{- (2^{N B C - 1} + 1)} \leq x < 2^{2^{N B C - 1} - 1}

	NBC = 8
2¹²⁷ =	0.17014 11834 60469 23173 16873 03715 88410(39)
2⁻¹²⁹ =	.14693 67938 52785 93849 60920 67152 78070(−38)
log_e(2¹²⁷) =	88.02969 19311 13054 29598 84794 25188 42414
log_e(2⁻¹²⁹) =	−89.41598 62922 32944 91482 29436 68104 77728
	NBC = 11
2¹⁰²³ =	0.89884 65674 31157 95386 46525 95394 51236(308)
2⁻¹⁰²⁵ =	.27813 42323 13400 17288 62790 89666 55050(−308)
log_e(2¹⁰²³) =	709.08956 57128 24051 53382 84602 51714 62914
log_e(2⁻¹⁰²⁵) =	−710.47586 00739 43942 15266 29244 94630 98227

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TABLE 1.

ERFC(Z)

RHO\THETA	0°		15°		30°		37.5°		45°
.00	.1000004+01	.0	.100000+01	.000000	.100000+01	.000000	.100000+01	.000000	.100000+01	.000000
.02	.977435+00	.0	.978204+00	−.583879−02	.980456+00	−.112808−01	.982095+00	−.137355−01	.984040+00	−.159556−01
.04	.954889+00	.0	.956420+00	−.116648−01	.960912+00	−.225435−01	.964183+00	−.274543−01	.968068+00	−.318984−01
.06	.932378+00	.0	.934662+00	−.174654−01	.941368+00	−.337702−01	.946257+00	−.411398−01	.952070+00	−.478156−01
.08	.909922+00	.0	.912942+00	−.232279−01	.921824+00	−.449428−01	.928310+00	−.547751−01	.936033+00	−.636943−01
.10	.887537+00	.0	.891273+00	−.289397−01	.902280+00	−.560434−01	.910337+00	−.683437−01	.919946+00	−.795217−01
.12	.865242+00	.0	.869667+00	−.345886−01	.882738+00	−.670542−01	.892330+00	−.818288−01	.903796+00	−.952846−01
.14	.843053+00	.0	.848138+00	−.401625−01	.863197+00	−.779575−01	.874283+00	−.952136−01	.887571+00	−.110970+00
.16	.820988+00	.0	.826697+00	−.456493−01	.843657+00	−.887357−01	.856190+00	−.108481+00	.871258+00	−.126564+00
.18	.799064+00	.0	.805358+00	−.510376−01	.824121+00	−.993713−01	.838045+00	−.121615+00	.854845+00	−.142053+00
.20	.777297+00	.0	.784132+00	−.563161−01	.804590+00	−.109847+00	.819844+00	−.134598+00	.838321+00	−.157424+00
.30	.671373+00	.0	.680129+00	−.806921−01	.707071+00	−.159241+00	.727824+00	−.196662+00	.753652+00	−.231995+00
.40	.571608+00	.0	.580740+00	−.100871+00	.610081+00	−.202202+00	.633845+00	−.252418+00	.664672+00	−.301347+00
.50	.479500+00	.0	.487320+00	−.115988+00	.514270+00	−.236937+00	.537866+00	−.299764+00	.570447+00	−.363359+00
.60	.396144+00	.0	.401058+00	−.125565+00	.420627+00	−.261983+00	.440342+00	−.336667+00	.470518+00	−.415645+00
.70	.322199+00	.0	.322926+00	−.129533+00	.330478+00	−.276308+00	.342317+00	−.361274+00	.365070+00	−.455585+00
.80	.257899+00	.0	.253631+00	−.128213+00	.245436+00	−.279418+00	.245465+00	−.372052+00	.255106+00	−.480414+00
.90	.203092+00	.0	.193577+00	−.122258+00	.167296+00	−.271437+00	.152078+00	−.367979+00	.142621+00	−.487404+00
1.00	.157299+00	.0	.142847+00	−.112577+00	.978859−01	−.253165+00	.649787−01	−.348749+00	.307358−01	−.474148+00
1.10	.119795+00	.0	.101201+00	−.100228+00	.388720−01	−.226077+00	−.126659−01	−.314972+00	−.762502−01	−.438939+00
1.20	.896860−01	.0	.680972−01	−.863111−01	−.844458−02	−.192259+00	−.776217−01	−.268341+00	−.172938+00	−.381252+00
1.30	.659921−01	.0	.427427−01	−.718743−01	−.433244−01	−.154271+00	−.126958+00	−.211694+00	−.253111+00	−.302268+00
1.40	.477149−01	.0	.241557−01	−.578252−01	−.657413−01	−.114934+00	−.158478+00	−.148963+00	−.310259+00	−.205369+00
1.50	.338949−01	.0	.112438−01	−.448760−01	−.764609−01	−.770708−01	−.171150+00	−.849352−01	−.338390+00	−.965018−01
1.60	.236516−01	.0	.288326−02	−.335145−01	−.770202−01	−.432136−01	−.165475+00	−.248347−01	−.333094+00	.157600−01
1.70	.162095−01	.0	−.200643−02	−.240031−01	−.696000−01	−.153368−01	−.143694+00	.262601−01	−.292775+00	.120601+00
1.80	.109095−01	.0	−.440212−02	−.164025−01	−.567976−01	.535338−02	−.109748+00	.641075−01	−.219843+00	.205946+00
1.90	.720957−02	.0	−.513475−02	−.106123−01	−.413301−01	.185261−01	−.689521−01	.859866−01	−.121580+00	.260096+00
2.00	.467773−02	.0	−.486777−02	−.641967−02	−.257133−01	.247077−01	−.273484−01	.912368−01	−.103117−01	.273926+00
2.10	.297947−02	.0	−.409504−02	−.354979−02	−.119733−01	.251347−01	.916428−02	.815039−01	.974865−01	.243355+00
2.20	.186285−02	.0	−.315468−02	−.170957−02	−.144281−02	.215037−01	.358034−01	.605788−01	.183390+00	.171536+00
2.30	.114318−02	.0	−.225377−02	−.621965−03	.531949−02	.156622−01	.497620−01	.337876−01	.230486+00	.700421−01
2.40	.688514−03	.0	−.149843−02	−.481389−04	.847425−02	.930477−02	.507367−01	.700451−02	.227631+00	−.417236−01
2.50	.406952−03	.0	−.924217−03	.201683−03	.872881−02	.372996−02	.409438−01	−.145264−01	.173531+00	−.139462+00
2.60	.236034−03	.0	−.523179−03	.266848−03	.709056−02	−.293708−03	.245557−01	−.272819−01	.792740−01	−.199249+00
2.70	.134333−03	.0	−.265334−03	.241454−03	.460569−02	−.253794−02	.664754−02	−.301636−01	−.321336−01	−.204295+00
2.80	.750132−04	.0	−.113783−03	.183127−03	.214179−02	−.321372−02	−.809354−02	−.245881−01	−.130194+00	−.151392+00
2.90	.410979−04	.0	−.339435−04	.122861−03	.256793−03	−.279917−02	−.165397−01	−.138997−01	−.185124+00	−.545715−01
3.00	.220905−04	.0	.198248−05	.741051−04	−.830403−03	−.184961−02	−.178141−01	−.225516−02	−.178018+00	.564096−01
3.50	.743098−06	.0	.383634−05	−.331302−06	.361438−04	.343118−03	.663124−02	.774278−03	.146013+00	−.667835−01
4.00	.154173−07	.0	−.506480−07	−.121829−06	−.100583−04	−.454411−04	−.206887−02	.808579−03	−.704373−01	.121816+00
4.50	.196616−09	.0	−.172664−08	.241877−08	.342245−05	.358814−05	.149447−03	−.641621−03	−.697054−01	−.103987+00
5.00	.153746−11	.0	.431344−10	−.812606−11	−.410390−06	.690194−07	.163226−03	.593711−04	.909031−01	−.666628−01
5.50	.735785−14	.0	−.401550−12	−.137042−12	−.968939−10	−.274562−07	.529439−06	.406179−04	.945377−01	.396363−01
6.00	.215197−16	.0	.222076−14	.150084−14	.136924−08	−.383336−09	−.550781−05	.636094−05	.563713−01	.752047−01
6.50	.384215−19	.0	−.905982−17	−.633184−17	.477805−10	.324007−10	−.126107−05	.886874−06	.498547−01	.710156−01
7.00	.418383−22	.0	.276523−19	.109112−19	.941894−12	.157585−11	−.164519−06	.187753−06	.710858−01	.379415−01
7.50	.277665−25	.0	−.517996−22	.562585−23	.233092−13	.393021−13	.525818−10	.356932−07	.667858−01	−.345871−01
8.00	.112243−28	.0	.396093−25	−.443759−25	.731135−15	.506747−15	.420606−08	.160997−08	−.258232−01	−.656143−01
8.50	.276232−32	.0	.111560−28	.427748−28	.130972−16	−.343079−17	.123912−09	−.485650−09	−.475602−01	.462887−01
9.00	.413703−36	.0	−.213063−31	−.174915−32	.313042−20	−.160996−18	−.462157−10	.168052−10	.623821−01	−.612171−02
RHO\THETA	50°		60°		70°		80°		90°
.00	.100000+01	.000000	.100000+01	.000000	.100000+01	.000000	.100000+01	.000000	.1+01	.000000
.02	.985491+00	−.172863−01	.988713+00	−.195441−01	.992279+00	−.212081−01	.996080+00	−.222273−01	.1+01	−.225706−01
.04	.970967+00	−.345635−01	.977408+00	−.390882−01	.984542+00	−.424252−01	.992150+00	−.444703−01	.1+01	−.451593−01
.06	.956411+00	−.518226−01	.966067+00	−.586322−01	.976774+00	−.636604−01	.988203+00	−.667446−01	.1+01	−.677841−01
.08	.941809+00	−.690545−01	.954672+00	−.781761−01	.968959+00	−.849226−01	.984228+00	−.890659−01	.1+01	−.904633−01
.10	.927144+00	−.862497−01	.943204+00	−.977195−01	.961080+00	−.106221+00	.980217+00	−.111450+00	.1+01	−.113215+00
.12	.912401+00	−.103399+00	.931646+00	−.117262+00	.953123+00	−.127564+00	.976160+00	−.133913+00	.1+01	−.136058+00
.14	.897565+00	−.120493+00	.919978+00	−.136803+00	.945070+00	−.148961+00	.972048+00	−.156471+00	.1+01	−.159011+00
.16	.882621+00	−.137521+00	.908183+00	−.156342+00	.936906+00	−.170421+00	.967870+00	−.179140+00	.1+01	−.182093+00
.18	.867552+00	−.154473+00	.896242+00	−.175878+00	.928612+00	−.191952+00	.963617+00	−.201936+00	.1+01	−.205323+00
.20	.852345+00	−.171339+00	.884135+00	−.195409+00	.920173+00	−.213564+00	.959279+00	−.224877+00	.1+01	−.228721+00
.30	.773712+00	−.253982+00	.820454+00	−.292919+00	.875153+00	−.323124+00	.935924+00	−.342344+00	.1+01	−.348949+00
.40	.689468+00	−.332643+00	.749698+00	−.389843+00	.823615+00	−.435932+00	.908660+00	−.466099+00	.1+01	−.476625+00
.50	.598041+00	−.405420+00	.669242+00	−.485367+00	.762712+00	−.552889+00	.875613+00	−.598673+00	.1+01	−.614952+00
.60	.498195+00	−.469950+00	.576286+00	−.578076+00	.688962+00	−.674665+00	.834330+00	−.742997+00	.1+01	−.767853+00
.70	.389205+00	−.523342+00	.467898+00	−.665719+00	.598036+00	−.801524+00	.781501+00	−.902533+00	.1+01	−.940283+00
.80	.271098+00	−.562163+00	.341135+00	−.744933+00	.484501+00	−.933057+00	.712565+00	−.108143+01	.1+01	−.113867+01
.90	.144933+00	−.582512+00	.193283+00	−.810912+00	.341532+00	−.106775+01	.621138+00	−.128468+01	.1+01	−.137154+01
1.00	.131385−01	−.580211+00	.222736−01	−.857061+00	.160625+00	−.120233+01	.498184+00	−.151831+01	.1+01	−.165043+01
1.10	−.120165+00	−.551166+00	−.172655+00	−.874670+00	−.686592−01	−.133079+01	.330784+00	−.178947+01	.1+01	−.199117+01
1.20	−.248882+00	−.491928+00	−.389989+00	−.852699+00	−.358812+00	−.144298+01	.100318+00	−.210652+01	.1+01	−.241591+01
1.30	−.364829+00	−.400481+00	−.624572+00	−.777828+00	−.724161+00	−.152248+01	−.220245+00	−.247879+01	.1+01	−.295609+01
1.40	−.457890+00	−.277246+00	−.865837+00	−.635009+00	−.118000+01	−.154380+01	−.670390+00	−.291589+01	.1+01	−.365696+01
1.50	−.516623+00	−.126215+00	−.109565+01	−.408856+00	−.174051+01	−.146840+01	−.130809+01	−.342593+01	.1+01	−.458473+01
1.60	−.529455+00	.439353−01	−.128600+01	−.863300−01	−.241452+01	−.123975+01	−.221880+01	−.401186+01	.1+01	−.583773+01
1.70	−.486589+00	.219059+00	−.139708+01	.338770+00	−.319777+01	−.777391+00	−.352885+01	−.466405+01	.1+01	−.756418+01
1.80	−.382619+00	.379468+00	−.137659+01	.859011+00	−.405945+01	.291660−01	−.542509+01	−.534607+01	.1+01	−.999112+01
1.90	−.219614+00	.501081+00	−.116196+01	.144402+01	−.492009+01	.132454+01	−.818351+01	−.596797+01	.1+01	−.134718+02
2.00	−.101646−01	.558267+00	−.687539+00	.202914+01	−.561710+01	.328529+01	−.122098+02	−.633691+01	.1+01	−.185648+02
2.10	.220530+00	.526666+00	.100155+00	.250402+01	−.585370+01	.610278+01	−.180944+02	−.606686+01	.1+01	−.261677+02
2.20	.434783+00	.399806+00	.121649+01	.270506+01	−.512891+01	.993582+01	−.266833+02	−.441546+01	.1+01	−.377471+02
2.30	.586112+00	.176582+00	.260024+01	.241951+01	−.265168+01	.148074+02	−.391538+02	.862691−02	.1+01	−.557397+02
2.40	.627560+00	−.112358+00	.406480+01	.141207+01	.274334+01	.204062+02	−.570679+02	.976572+01	.1+01	−.842631+02
2.50	.524677+00	−.412231+00	.524907+01	−.511462+00	.126293+02	.257378+02	−.823162+02	.294995+02	.1+01	−.130396+03
2.60	.271303+00	−.647141+00	.559076+01	−.339050+01	.288616+02	.285642+02	−.116747+03	.675244+02	.1+01	−.206519+03
2.70	−.961516−01	−.735197+00	.436233+01	−.694257+01	.530807+02	.245883+02	−.161012+03	.138532+03	.1+01	−.334671+03
2.80	−.492661+00	−.613886+00	.827238+00	−.103663+02	.854677+02	.644245+01	−.211587+03	.268092+03	.1+01	−.554777+03
2.90	−.794447+00	−.271241+00	−.542122+01	−.121876+02	.122167+03	−.371981+02	−.253771+03	.499853+03	.1+01	−.940470+03
3.00	−.870118+00	.227570+00	−.139065+02	−.102944+02	.150602+03	−.121463+03	−.246068+03	.906353+03	.1+01	−.162999+04
3.50	.128375+01	−.438817+00	.436770+02	.609546+02	−.185185+04	−.696951+03	.126726+05	.110011+05	.1+01	−.352823+05
4.00	−.143233+01	.176970+01	−.286672+03	−.315634+03	.142504+05	.268505+05	.414164+06	−.267047+06	.1+01	−.129696+07
4.50	−.153046+01	−.394580+01	.303475+04	.899044+03	−.581405+05	−.694896+06	−.104472+08	−.211627+08	.1+01	−.801975+08
5.00	.818979+01	−.289861+01	−.236016+05	.194203+05	−.301677+05	.237891+08	−.159723+10	.902680+09	.1+01	−.829827+10
5.50	.141149+02	.136673+02	−.192298+06	−.331209+06	−.279467+09	−.117057+10	.155714+12	.170313+12	.1+01	−.143210+13
6.00	.892289+01	.480481+02	.434927+07	−.444020+07	.640472+11	.633707+11	.196750+14	−.425327+14	.1+01	−.411275+15
6.50	.767743+01	.133290+03	.130111+09	.950077+07	−.992019+13	.967322+12	−.152018+17	.205488+16	.1+01	−.196225+18
7.00	.164446+03	.364872+03	.308919+10	.172358+10	.442269+15	−.156615+16	.619376+19	.519553+19	.1+01	−.155349+21
7.50	.126058+04	.375323+03	.107926+12	.606848+11	.371118+18	.125665+18	−.143772+22	−.670033+22	.1+01	−.203882+24
8.00	.231199+04	−.413300+04	.557097+13	.462008+12	−.785527+19	.138783+21	−.257275+25	.897512+25	.1+01	−.443245+27
8.50	−.181429+05	−.441024+04	.231401+15	−.228725+15	−.624892+23	.370071+23	.115166+29	−.168785+29	.1+01	−.159530+31
9.00	.404091+05	.697050+05	−.117844+17	−.213680+17	−.554262+26	−.688648+25	−.488030+32	.526592+32	.1+01	−.950078+34

Open in a new tab

TABLE 2.

EXP(Z**2)*ERFC(Z)

RHO\THETA	0°		15°		30°		37.5°		45°
.00	.100000+01	.0	.100000+01	.000000	.100000+01	.000000	.100000+01	.000000	.100000+01	.000000
.02	.977826+00	.0	.978544+00	−.564511−02	.980656+00	−.109433−01	.982202+00	−.133574−01	.984046+00	−.155619−01
.04	.956418+00	.0	.957755+00	−.109148−01	.961711+00	−.212290−01	.964623+00	−.259749−01	.968117+00	−.303494−01
.06	.935741+00	.0	.937610+00	−.158323−01	.943165+00	−.308907−01	.947276+00	−.378843−01	.952236+00	−.443878−01
.08	.915764+00	.0	.918086+00	−.204192−01	.925014+00	−.399605−01	.930171+00	−.491167−01	.936422+00	−.577025−01
.10	.896457+00	.0	.899159+00	−.246960−01	.907257+00	−.484692−01	.913315+00	−.597018−01	.920696+00	−.703184−01
.12	.877791+00	.0	.880810+00	−.286817−01	.889890+00	−.564458−01	.896717+00	−.696688−01	.905075+00	−.822605−01
.14	.859740+00	.0	.863016+00	−.323942−01	.872908+00	−.639183−01	.880383+00	−.790455−01	.889575+00	−.935532−01
.16	.842277+00	.0	.845758+00	−.358502−01	.856308+00	−.709132−01	.864318+00	−.878590−01	.874212+00	−.104221+00
.18	.825378+00	.0	.829017+00	−.390656−01	.840085+00	−.774557−01	.848526+00	−.961353−01	.858998+00	−.114286+00
.20	.809020+00	.0	.812775+00	−.420551−01	.824233+00	−.835698−01	.833010+00	−.103900+00	.843946+00	−.123774+00
.30	.734599+00	.0	.738441+00	−.540691−01	.750340+00	−.108476+00	.759640+00	−.135855+00	.771453+00	−.163319+00
.40	.670788+00	.0	.674177+00	−.621855−01	.684812+00	−.125660+00	.693269+00	−.158259+00	.704192+00	−.191604+00
.50	.615690+00	.0	.618355+00	−.674592−01	.626815+00	−.137032+00	.633647+00	−.173298+00	.642609+00	−.210932+00
.60	.567805+00	.0	.569635+00	−.706620−01	.575515+00	−.144060+00	.580342+00	−.182718+00	.586778+00	−.223250+00
.70	.525930+00	.0	.526911+00	−.723596−01	.530125+00	−.147863+00	.532829+00	−.187900+00	.536524+00	−.230166+00
.80	.489101+00	.0	.489272+00	−.729681−01	.489917+00	−.149296+00	.490547+00	−.189926+00	.491521+00	−.232989+00
.90	.456532+00	.0	.455962+00	−.727939−01	.454239+00	−.149005+00	.452940+00	−.189631+00	.451358+00	−.232766+00
1.00	.427584+00	.0	.426354+00	−.720633−01	.422513+00	−.147478+00	.419480+00	−.187662+00	.415588+00	−.230320+00
1.10	.401730+00	.0	.399923+00	−.709434−01	.394231+00	−.145081+00	.389679+00	−.184510+00	.383760+00	−.226295+00
1.20	.378537+00	.0	.376233+00	−.695576−01	.368949+00	−.142090+00	.363094+00	−.180548+00	.355440+00	−.221186+00
1.30	.357643+00	.0	.354916+00	−.679971−01	.346284+00	−.138710+00	.339332+00	−.176056+00	.330223+00	−.215369+00
1.40	.338744+00	.0	.335662+00	−.663291−01	.325905+00	−.135093+00	.318044+00	−.171244+00	.307738+00	−.209128+00
1.50	.321585+00	.0	.318209+00	−.646029−01	.307525+00	−.131350+00	.298923+00	−.166267+00	.287653+00	−.202672+00
1.60	.305953+00	.0	.302334+00	−.628545−01	.290896+00	−.127564+00	.281701+00	−.161236+00	.269672+00	−.196156+00
1.70	.291663+00	.0	.287848+00	−.611100−01	.275808+00	−.123794+00	.266147+00	−.156235+00	.253535+00	−.189690+00
1.80	.278560+00	.0	.274589+00	−.593879−01	.262075+00	−.120081+00	.252058+00	−.151319+00	.239013+00	−.183351+00
1.90	.266509+00	.0	.262415+00	−.577015−01	.249540+00	−.116455+00	.239260+00	−.146528+00	.225908+00	−.177191+00
2.00	.255396+00	.0	.251208+00	−.560596−01	.238065+00	−.112935+00	.227600+00	−.141889+00	.214048+00	−.171246+00
2.10	.245119+00	.0	.240863+00	−.544681−01	.227532+00	−.109534+00	.216948+00	−.137417+00	.203282+00	−.165535+00
2.20	.235593+00	.0	.231289+00	−.529306−01	.217837+00	−.106258+00	.207188+00	−.133122+00	.193480+00	−.160068+00
2.30	.226742+00	.0	.222408+00	−.514490−01	.208891+00	−.103112+00	.198222+00	−.129008+00	.184530+00	−.154849+00
2.40	.218499+00	.0	.214150+00	−.500239−01	.200615+00	−.100095+00	.189963+00	−.125073+00	.176334+00	−.149876+00
2.50	.210806+00	.0	.206456+00	.486549−01	.192941+00	−.972068−01	.182336+00	−.121315+00	.168808+00	−.145143+00
2.60	.203613+00	.0	.199271+00	−.473413−01	.185809+00	−.944437−01	.175275+00	−.117730+00	.161878+00	−.140642+00
2.70	.196874+00	.0	.192549+00	−.460815−01	.179166+00	−.918020−01	.168723+00	−.114311+00	.155479+00	−.136364+00
2.80	.190549+00	.0	.186248+00	−.448740−01	.172967+00	−.892774−01	.162630+00	−.111051+00	.149556+00	−.132298+00
2.90	.184602+00	.0	.180331+00	−.437168−01	.167168+00	−.868650−01	.156950+00	−.107943+00	.144061+00	−.128434+00
3.00	.179001+00	.0	.174766+00	−.426079−01	.161735+00	−.845599−01	.151645+00	−.104980+00	.138950+00	−.124761+00
3.50	.155294+00	.0	.151281+00	−.377161−01	.139036+00	−.744669−01	.129658+00	−.920841−01	.117989+00	−.108897+00
4.00	.136999+00	.0	.133241+00	−.337346−01	.121846+00	−.663430−01	.113196+00	−.817946−01	.102526+00	−.963790−01
4.50	.122485+00	.0	.118978+00	−.304579−01	.108404+00	−.597166−01	.100432+00	−.734593−01	.906646−01	−.863253−01
5.00	.110705+00	.0	.107435+00	−.277278−01	.976147−01	−.542350−01	.902530−01	−.666012−01	.812804−01	−.781076−01
5.50	.100962+00	.0	.979088−01	−.254257−01	.887699−01	−.496392−01	.819492−01	−.608758−01	.736702−01	−.712820−01
6.00	.927766−01	.0	.899189−01	−.234628−01	.813899−01	−.457386−01	.750469−01	−.560327−01	.673730−01	−.655313−01
6.50	.858057−01	.0	.831244−01	−.217720−01	.751399−01	−.423911−01	.692192−01	−.518878−01	.620749−01	−.606252−01
7.00	.798001−01	.0	.772774−01	−.203020−01	.697797−01	−.394900−01	.642331−01	−.483034−01	.575547−01	−.563934−01
7.50	.745737−01	.0	.721940−01	−.190134−01	.651322−01	−.369532−01	.599186−01	−.451749−01	.536520−01	−.527075−01
8.00	.699852−01	.0	.677345−01	−.178753−01	.610646−01	−.347176−01	.561483−01	−.424219−01	.502479−01	−.494694−01
8.50	.659251−01	.0	.637913−01	−.168633−01	.574747−01	−.327332−01	.528254−01	−.399814−01	.472522−01	−.466031−01
9.00	.623077−01	.0	.602800−01	−.159579−01	.542832−01	−.309606−01	.498745−01	−.378036−01	.445953−01	−.440484−01
RHO\THETA	50°		60°		70°		80°		90°
.00	.100000+01	.000000	.100000+01	.000000	.100000+01	.000000	.100000+01	.000000	.100000+01	.000000
.02	.985430+00	−.168969−01	.988522+00	−.191978−01	.991980+00	−.209465−01	.995708+00	−.220828−01	.999600+00	−.225616−01
.04	.970750+00	−.330243−01	.976680+00	−.377036−01	.983379+00	−.413619−01	.990684+00	−.438614−01	.998401+00	−.450871−01
.06	.955991+00	−.484013−01	.964508+00	−.555200−01	.974228+00	−.612308−01	.984947+00	−.653065−01	.996406+00	−.675405−01
.08	.941179+00	−.630470−01	.952039+00	−.726507−01	.964559+00	−.805399−01	.978518+00	−.863902−01	.993620+00	−.898862−01
.10	.926338+00	−.769813−01	.939307+00	−.891010−01	.954404+00	−.992775−01	.971421+00	−.107086+00	.990050+00	−.112089+00
.12	.911494+00	−.902242−01	.926342+00	−.104877+00	.943794+00	−.117434+00	.963679+00	−.127369+00	.985703+00	−.134113+00
.14	.896667+00	−.102796+00	.913175+00	−.119987+00	.932761+00	−.135001+00	.955317+00	−.147215+00	.980591+00	−.155925+00
.16	.881878+00	−.114717+00	.899834+00	−.134439+00	.921336+00	−.151972+00	.946360+00	−.166602+00	.974725+00	−.177491+00
.18	.867146+00	−.126007+00	.886348+00	−.148243+00	.909551+00	−.168342+00	.936834+00	−.185511+00	.968119+00	−.198777+00
.20	.852490+00	−.136687+00	.872742+00	−.161410+00	.897434+00	−.184108+00	.926768+00	−.203921+00	.960789+00	−.219753+00
.30	.780858+00	−.181638+00	.803774+00	−.218107+00	.832921+00	−.253864+00	.869300+00	−.287964+00	.913931+00	−.318916+00
.40	.713036+00	−.214294+00	.735131+00	−.260834+00	.764351+00	−.308806+00	.802581+00	−.357672+00	.852144+00	−.406153+00
.50	.649979+00	−.236926+00	.668830+00	−.291462+00	.694712+00	−.349876+00	.730181+00	−.412479+00	.778801+00	−.478925+00
.60	.592156+00	−.251548+00	.606260+00	−.311920+00	.626410+00	−.378484+00	.655426+00	−.452678+00	.697676+00	−.535713+00
.70	.539686+00	−.259892+00	.548281+00	−.324060+00	.561255+00	−.396308+00	.581214+00	−.479263+00	.612626+00	−.576042+00
.80	.492447+00	−.263412+00	.495329+00	−.329575+00	.500496+00	−.405125+00	.509895+00	−.493727+00	.527292+00	−.600412+00
.90	.450161+00	−.263301+00	.447520+00	−.329960+00	.444890+00	−.406682+00	.443222+00	−.497872+00	.444858+00	−.610142+00
1.00	.412460+00	−.260519+00	.404741+00	−.326488+00	.394786+00	−.402602+00	.382368+00	−.493623+00	.367879+00	−.607158+00
1.10	.378934+00	−.255830+00	.366725+00	−.320215+00	.350217+00	−.394324+00	.327976+00	−.482883+00	.298197+00	−.593761+00
1.20	.349161+00	−.249826+00	.333111+00	−.311994+00	.310985+00	−.383080+00	.280243+00	−.467415+00	.236928+00	−.572397+00
1.30	.322730+00	−.242965+00	.303494+00	−.302503+00	.276740+00	−.369883+00	.239021+00	−.448772+00	.184520+00	−.545456+00
1.40	.299256+00	−.235593+00	.277452+00	−.292262+00	.247038+00	−.355543+00	.203909+00	−.428253+00	.140858+00	−.515113+00
1.50	.278384+00	−.227969+00	.254575+00	−.281666+00	.221396+00	−.340682+00	.174350+00	−.406909+00	.105399+00	−.483227+00
1.60	.259793+00	−.220280+00	.234476+00	−.271004+00	.199324+00	−.325767+00	.149701+00	−.385528+00	.773047−01	−.451284+00
1.70	.243198+00	−.212662+00	.216798+00	−.260483+00	−180352+00	−.311128+00	.129300+00	−.364688+00	.555762−01	−.420388+00
1.80	.226349+00	−.205210+00	.201224+00	−.250247+00	.164045+00	−.296992+00	.112501+00	−.344773+00	.391639−01	−.391291+00
1.90	.215025+00	−.197986+00	.187470+00	−.240389+00	−150014+00	−.283502+00	.987098−01	−.326022+00	.270518−01	−.364437+00
2.00	.203034+00	−.191033+00	.175289+00	−.230965+00	.137912+00	−.270739+00	.873971−01	−.308553+00	.183156−01	−.340026+00
2.10	.192211+00	−.184372+00	.164467+00	−.222006+00	.127441+00	−.258737+00	.781048−01	−.292400+00	.121552−01	−.318073+00
2.20	.182411+00	−.178015+00	.154819+00	−.213521+00	−118345+00	−.247499+00	.704459−01	−.277541+00	.790705−02	−.298468+00
2.30	.173510+00	−.171964+00	.146185+00	−.205507+00	.110409+00	−.237007+00	.641001−01	−.263913+00	.504176−02	−.281026+00
2.40	.165400+00	−.166214+00	.138430+00	−.197951+00	.103450+00	−.227227+00	.588065−01	−.251431+00	.315111−02	−.265522+00
2.50	.157988+00	−.160758+00	.131439+00	−.190836+00	.973162−01	−.218118+00	.543547−01	−.240002+00	.193045−02	−.251723+00
2.60	.151195+00	−.155584+00	.125111+00	−.184138+00	.918812−01	−.209636+00	.505775−01	−.229528+00	.115923−02	−.239403+00
2.70	.144950+00	−.150680+00	.119364+00	−.177833+00	.870395−01	−.201733+00	.473424−01	−.219915+00	.682328−03	−.228355+00
2.80	.139193+00	−.146031+00	.114124+00	−.171899+00	.827037−01	−.194364+00	.445450−01	−.211075+00	.393669−03	−.218399+00
2.90	.133871+00	−.141624+00	.109330+00	−.166309+00	.788013−01	−.187486+00	.421033−01	−.202926+00	.222630−03	−.209377+00
3.00	.128940+00	−.137445+00	.104930+00	−.161041+00	.752717−01	−.181058+00	.399530−01	−.195396+00	.123410−03	−.201157+00
3.50	.108892+00	−.119509+00	.874505−01	−.138780+00	.617088−01	−.154443+00	.321214−01	−.165001+00	.478512−05	−.168830+00
4.00	.942797−01	−.105481+00	.750909−01	−.121741+00	.525024−01	−.134607+00	.270928−01	−.143006+00	.112535−06	−.145954+00
4.50	.831655−01	−.942896−01	.658787−01	−.108356+00	.458048−01	−.119295+00	.235275−01	−.126308+00	.160523−08	−.128735+00
5.00	.744259−01	−.851875−01	.587346−01	−.975919−01	.406890−01	−.107125+00	.208412−01	−.113170+00	.138879−10	−.115246+00
5.50	.673700−01	−.776560−01	.530232−01	−.887581−01	.366399−01	−.972205−01	.187325−01	−.102546+00	.728772−13	−.104367+00
6.00	.615511−01	−.713293−01	.483467−01	−.813834−01	.333477−01	−.890010−01	.170271−01	−.937703−01	.231952−15	−.953962−01
6.50	.566679−01	−.659441−01	.444434−01	−.751362−01	.306137−01	−.820692−01	.156161−01	−.863935−01	.447773−18	−.878644−01
7.00	.525101−01	−.613075−01	.411335−01	−.697775−01	.283042−01	−.761436−01	.144276−01	−.801029−01	.524289−21	−.814475−01
7.50	.489261−01	−.572750−01	.382898−01	−.651309−01	.263257−01	−.710192−01	.134115−01	−.746733−01	.372336−24	−.759126−01
8.00	.458040−01	−.537368−01	.358190−01	−.610637−01	.246107−01	−.665434−01	.125322−01	−.699379−01	.160381−27	−.710881−01
8.50	.430594−01	−.506078−01	.336516−01	−.574741−01	.231091−01	−.626001−01	.117632−01	−.657709−01	.419009−31	−.668445−01
9.00	.406273−01	−.478214−01	.317343−01	−.542828−01	.217828−01	−.590993−01	.110848−01	−.620752−01	.663968−35	−.630821−01

Open in a new tab

TABLE 3.

EXP(Z**2)*ERFC(−Z)

RHO\THETA	0°		15°		30°		37.5°		45°
.00	.100000+01	.0	.100000+01	.000000	.100000+01	.000000	.100000+01	.000000	.100000+01	.000000
.02	.102297+01	.0	.102215+01	.604525−02	.101974+01	.116363−01	.101801+01	.141302−01	.101595+01	.163619−01
.04	.104678+01	.0	.104502+01	.125170−01	.103989+01	.240025−01	.103620+01	.290671−01	.103188+01	.335494−01
.06	.107147+01	.0	.106863+01	.194435−01	.106043+01	.371373−01	.105458+01	.448455−01	.104775+01	.515878−01
.08	.109708+01	.0	.109302+01	.268548−01	.108137+01	.510811−01	.107311+01	.615009−01	.106354+01	.705024−01
.10	.112364+01	.0	.111821+01	.347829−01	.110269+01	.658763−01	.109177+01	.790701−01	.107920+01	.903181−01
.12	.115122+01	.0	.114424+01	.432622−01	.112441+01	.815669−01	.111056+01	.975904−01	.109472+01	.111059+00
.14	.117985+01	.0	.117112+01	.523294−01	.114650+01	.981992−01	.112943+01	.117100+00	.111004+01	.132751+00
.16	.120958+01	.0	.119891+01	.620234−01	.116896+01	.115821+00	.114836+01	.137638+00	.112513+01	.155415+00
.18	.124048+01	.0	.122763+01	.723862−01	.119178+01	.134483+00	.116733+01	.159244+00	.113995+01	.179075+00
.20	.127260+01	.0	.125731+01	.834622−01	.121495+01	.154237+00	.118629+01	.181958+00	.115445+01	.203753+00
.30	.145375+01	.0	.142149+01	.151332+00	.133536+01	.271371+00	.127976+01	.313595+00	.122045+01	.343076+00
.40	.167623+01	.0	.161572+01	.245769+00	.146100+01	.424910+00	.136645+01	.479143+00	.127026+01	.510240+00
.50	.195236+01	.0	.184573+01	.377084+00	.158657+01	.623876+00	.143813+01	.683551+00	.129522+01	.705740+00
.60	.229885+01	.0	.211791+01	.559713+00	.170349+01	.878536+00	.148357+01	.930808+00	.128502+01	.927798+00
.70	.273870+01	.0	.243899+01	.813903+00	.179848+01	.119994+01	.148800+01	.122283+01	.122814+01	.117142+01
.80	.330386+01	.0	.281529+01	.116807+01	.185201+01	.159889+01	.143293+01	.155787+01	.111267+01	.142738+01
.90	.403928+01	.0	.325121+01	.166206+01	.183636+01	.208415+01	.129634+01	.192846+01	.927639+00	.168134+01
1.00	.500898+01	.0	.374645+01	.235168+01	.171377+01	.265934+01	.105379+01	.231879+01	.665017+00	.191326+01
1.10	.630524+01	.0	.429102+01	.331474+01	.143482+01	.331817+01	.680682+00	.270192+01	.322278+00	.209753+01
1.20	.806285+01	.0	.485657+01	.465908+01	.938062+00	.403754+01	.156290+00	.303701+01	−.945930−01	.220410+01
1.30	.104813+02	.0	.538160+01	.653258+01	.151941+00	.476793+01	−.530066+00	.326754+01	−.568066+00	.220118+01
1.40	.138599+02	.0	.574679+01	.913501+01	−.998816+00	.542135+01	−.137052+01	.332164+01	−.106664+01	.205955+01
1.50	.186539+02	.0	.573426+01	.127298+02	−.257974+01	.585743+01	−.232809+01	.311621+01	−.154400+01	.175882+01
1.60	.255657+02	.0	.496175+01	.176520+02	−.462254+01	.587039+01	−.332543+01	.256681+01	−.194085+01	.129487+01
1.70	.356950+02	.0	.277771+01	.243017+02	−.708673+01	.518191+01	−.423536+01	.160541+01	−.219057+01	.687583+00
1.80	.507889+02	.0	−.190182+01	.331040+02	−.980423+01	.344908+01	−.487786+01	.206798+00	−.222934+01	−.131466−01
1.90	.736656+02	.0	−.108404+02	.443947+02	−.124081+02	.301777+00	−.502963+01	−.157716+01	−.201049+01	−.725740+00
2.00	.108941+03	.0	−.268411+02	.581561+02	−.142543+02	−.457095+01	−.445375+01	−.358054+01	−.152134+01	−.134236+01
2.10	.164294+03	.0	−.542404+02	.734660+02	−.143607+02	−.112629+02	−.295578+01	−.549415+01	−.798885+00	−.174372+01
2.20	.252703+03	.0	−.995202+02	.874162+02	−.114096+02	−.194032+02	−.468263+00	−.686146+01	.610500−01	−.182367+01
2.30	.396460+03	.0	−.171913+03	.930880+02	−.389141+01	−.278220+02	.284504+01	−.712231+01	.907518+00	−.152069+01
2.40	.634478+03	.0	−.283609+03	.759226+02	.950566+01	−.341808+02	.649023+01	−.572745+01	.155613+01	−.849408+00
2.50	.103581+04	.0	−.448599+03	.748935+01	.291411+02	−.347103+02	.959591+01	−.233541+01	.183009+01	.787847−01
2.60	.172508+04	.0	−.677955+03	−.164667+03	.532363+02	−.243320+02	.109818+02	.292473+01	.161505+01	.105855+01
2.70	.293094+04	.0	−.967019+03	−.532419+03	.763516+02	.239913+01	.941085+01	.919036+01	.913632+00	.182663+01
2.80	.508022+04	.0	−.126560+04	−.124780+04	.879744+02	.489857+02	.405857+01	.147289+02	−.121594+00	.213210+01
2.90	.898334+04	.0	−.141486+04	−.254465+04	.722456+02	.112885+03	−.485116+01	.171054+02	−.119968+01	.182716+01
3.00	.162060+05	.0	−.102319+04	−.474402+04	.105895+02	.179798+03	−.154399+02	.138266+02	−.196121+01	.948998+00
3.50	.417962+06	.0	.799689+05	−.127565+05	−.345002+03	−.846681+03	.352504+02	−.318118+02	.178275+01	−.513342+00
4.00	.177722+08	.0	−.303148+06	.206132+07	.165184+04	.572854+04	−.121850+03	.315751+02	−.201785+01	−.479428+00
4.50	.124593+10	.0	−.632047+08	−.532606+08	.127488+05	−.482630+05	.286262+03	.246430+03	.248394+00	.205738+01
5.00	.144010+12	.0	.504457+10	−.335304+09	−.505861+06	.179234+06	.714413+03	−.107588+04	.190113+01	−.186596+00
5.50	.274434+14	.0	−.398070+12	.262492+12	.359254+07	.647923+07	−.294447+04	−.407345+04	.714128+00	−.176703+01
6.00	.862246+16	.0	.457872+14	−.520744+14	.127587+09	−.310891+08	−.217456+05	−.476709+04	−.323300+00	−.191803+01
6.50	.446655+19	.0	−.100716+17	.118452+17	.133025+10	−.267648+10	−.112172+06	.340422+04	−.383674+00	−.191335+01
7.00	.381469+22	.0	.433489+19	−.317887+19	.207375+10	−.873217+11	−.630212+06	−.132005+06	.543630+00	−.185111+01
7.50	.537149+25	.0	−.283374+22	.426347+21	.630673+11	−.327704+13	−.252634+07	−.336114+07	.185781+01	−.535787+00
8.00	.124703+29	.0	.196497+25	.129886+25	.683728+14	−.142358+15	.165524+08	−.265076+08	.733467+00	.188952+01
8.50	.477316+32	.0	−.989764+25	−.298524+28	.943832+16	−.252614+16	.206701+09	.164806+09	−.204721+01	.598650−01
9.00	.301219+36	.0	−.549838+31	.194934+31	.397576+18	.666613+18	−.243169+10	.751742+09	.150878+01	−.121573+01
RHO\THETA	50°		60°		70°		80°		90°
.00	.100000+01	.000000	.100000+01	.000000	.100000+01	.000000	.100000+01	.000000	.100000+01	.000000
.02	.101443+01	.176847−01	.101108+01	.198904−01	.100741+01	.214606−01	.100354+01	.223563−01	.999600+00	.225616−01
.04	.102869+01	.361748−01	.102172+01	.404727−01	.101417+01	.434163−01	.100631+01	.449542−01	.998401+00	.450871−01
.06	.104275+01	.554874−01	.103189+01	.617442−01	.102026+01	.658462−01	.100830+01	.677607−01	.996406+00	.675405−01
.08	.105656+01	.756385−01	.104154+01	.837004−01	.102564+01	.887273−01	.100949+01	.907418−01	.993620+00	.898862−01
.10	.107009+01	.966430−01	.105064+01	.106335+00	.103029+01	.112035+00	.100986+01	.113862+00	.990050+00	.112089+00
.12	.108331+01	.118515+00	.105916+01	.129639+00	.103418+01	.135743+00	.100942+01	.137086+00	.985703+00	.134113+00
.14	.109617+01	.141267+00	.106704+01	.153603+00	.103728+01	.159822+00	.100814+01	.160377+00	.980591+00	.155925+00
.16	.110862+01	.164910+00	.107424+01	.178212+00	.103956+01	.184242+00	.100603+01	.183697+00	.974725+00	.177491+00
.18	.112062+01	.189453+00	.108074+01	.203453+00	.104100+01	.208971+00	.100307+01	.207009+00	.968119+00	.198777+00
.20	.113213+01	.214906+00	.108648+01	.229307+00	.104157+01	.233974+00	.999272+00	.230273+00	.960789+00	.219753+00
.30	.118040+01	.355926+00	.110242+01	.366982+00	.103071+01	.361797+00	.967640+00	.344526+00	.913931+00	.318916+00
.40	.120806+01	.519531+00	.109341+01	.515837+00	.995593+00	.490450+00	.915655+00	.451794+00	.852144+00	.406153+00
.50	.120731+01	.703662+00	.105496+01	.670616+00	.935432+00	.614113+00	.845306+00	.547520+00	.778801+00	.478925+00
.60	.116980+01	.903774+00	.983747+00	.824347+00	.851095+00	.726620+00	.759756+00	.627812+00	.697676+00	.535713+00
.70	.108743+01	.111228+01	.878285+00	.968587+00	.745231+00	.821976+00	.663105+00	.689772+00	.612626+00	.576042+00
.80	.953336+00	.131817+01	.739549+00	.109394+01	.622230+00	.894945+00	.560035+00	.731740+00	.527292+00	.600412+00
.90	.763292+00	.150697+0l	.571471+00	.119083+01	.487972+00	.941619+00	.455412+00	.753397+00	.444858+00	.610142+00
1.00	.517275+00	.166123+01	.381152+00	.125055+01	.349369+00	.959889+00	.353863+00	.755730+00	.367879+00	.607158+00
1.10	.221086+00	.176167+01	.178694+00	.126642+01	.213743+00	.949748+00	.259408+00	.740868+00	.298197+00	.593761+00
1.20	−.112292+00	.178923+01	−.234440−01	.123493+01	.881139−01	.913353+00	.175175+00	.711800+00	.236928+00	.572397+00
1.30	−.462011+00	.172780+01	−.211562+00	.115668+01	−.215044−0l	.854822+00	.103231+00	.672034+00	.184520+00	.545456+00
1.40	−.799794+00	.156770+01	−.372237+00	.103688+01	−.110704+00	.779790+00	.445420−01	.625237+00	.140858+00	.515113+00
1.50	−.109192+01	.130926+01	−.494065+00	.885190+00	−.177075+00	.694764+00	−.945684−03	.574901+00	.105399+00	.483227+00
1.60	−.130302+01	.965812+00	−.569332+00	.714952+00	−.220337+00	.606395+00	−.341317−01	.524075+00	.773047−01	.451284+00
1.70	−.140154+01	.565275+00	−.595323+00	.541594+00	−.242190+00	.520751+00	−.565267−01	.475193+00	.555762−01	.420388+00
1.80	−.136640+01	.149190+00	−.574932+00	.380624+00	−.245914+00	.442725+00	−.699973−01	.429992+00	.391639−01	.391291+00
1.90	−.119347+01	−.231426+00	−.516381+00	.245402+00	−.235801+00	.375650+00	−.765256−01	.389521+00	.270518−01	.364437+00
2.00	−.900424+00	−.523642+00	−.432005+00	.145177+00	−.216510+00	.321167+00	−.780102−01	.354223+00	.183156−01	.340026+00
2.10	−.527958+00	−.682837+00	−.336258+00	.837718−01	−.192467+00	.279344+00	−.761241−01	.324056+00	.121552−01	.318073+00
2.20	−.135761+00	−.683750+00	−.243313+00	.592582−01	−.167392+00	.248996+00	−.722347−01	.298640+00	.790705−02	.298468+00
2.30	.207215+00	−.529536+00	−.164751+00	.647151−01	−.144014+00	.228112+00	−.673774−01	.277393+00	.504176−02	.281026+00
2.40	.437243+00	−.255605+00	−.107845+00	.899283−01	−.123986+00	.214327+00	−.622737−01	.259650+00	.315111−02	.265522+00
2.50	.512073+00	.744257−01	−.748106−01	.123641+00	−.107986+00	.205321+00	−.573770−01	.244750+00	.193045−02	.251723+00
2.60	.424377+00	.381557+00	−.631827−01	.155822+00	−.959278−01	.199114+00	−.529309−01	.232099+00	.115923−02	.239403+00
2.70	.207358+00	.591072+00	−.671444−01	.179408+00	−.872383−01	.194224+00	−.490308−01	.221194+00	.682328−03	.228355+00
2.80	−.711724−01	.654101+00	−.794229−01	.191148+00	−.811203−01	.189696+00	−.456768−01	.211636+00	.393669−03	.218399+00
2.90	−.326684+00	.563991+00	−.932090−01	.191421+00	−.767654−01	.185036+00	−.428169−01	.203120+00	.222630−03	.209377+00
3.00	−.483674+00	.360592+00	−.103604+00	.183219+00	−.734911−01	.180089+00	−.403769−01	.195423+00	.123410−03	.201157+00
3.50	.999905−01	.472332−02	−.891007−01	.134728+00	−.617122−01	.154611+00	−.321314−01	.164984+00	.478512−05	.168830+00
4.00	−.218410+00	.993987−01	−.749050−01	.122386+00	−.525086−01	.134600+00	−.270923−01	.143006+00	.112535−06	.145954+00
4.50	−.558352−01	.147044+00	−.658583−01	.108279+00	−.458045−01	.119295+00	−.235275−01	.126308+00	.160523−08	.128735+00
5.00	−.517307−01	.724167−01	−.587417−01	.975944−01	−.406890−01	.107125+00	−.208412−01	.113170+00	.138879−10	.115246+00
5.50	−.679421−01	.672066−01	−.530230−01	.887586−01	−.366399−01	.972205−01	−.187325−01	.102546+00	.728772−13	.104367+00
6.00	−.639612−01	.683196−01	−.483467−01	.813834−01	−.333477−01	.890010−01	−.170271−01	.937703−01	.231952−15	.953962−01
6.50	−.576053−01	.650398−01	−.444434−01	.751362−01	−.306137−01	.820692−01	−.156161−01	.863935−01	.447773−18	.878644−01
7.00	−.526816−01	.609424−01	−.411335−01	.697775−01	−.283042−01	.761436−01	−.144276−01	.801029−01	.524289−21	.814475−01
7.50	−.488796−01	.571703−01	−.382898−01	.651309−01	−.263257−01	.710192−01	−.134115−01	.746733−01	.372336−24	.759126−01
8.00	−.457747−01	.537426−01	−.358190−01	.610637−01	−.246107−01	.665434−01	−.125322−01	.699379−01	.160381−27	.710881−01
8.50	−.430626−01	.506142−01	−.336516−01	.574741−01	−.231091−01	.626001−01	−.117632−01	.657709−01	.419009−31	.668445−01
9.00	−.406279−01	.478200−01	−.317343−01	.542828−01	−.217828−01	.590993−01	−.110848−01	.620752−01	.663968−35	.630821−01

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Footnotes

Part I. Error, Probability, and Related Functions. J. Res. Nat. Bur. Stand. (U.S.). 74(3): 211–224; 1970, Part II. The Exponential Integral E_n(x). J. Res. Nat. Bur. Stand. (U.S.). 78(4); 199–216; 1974. Part III. The Sine, Cosine, Exponential Integrals, and Related Functions. J. Res. Nat. Bur. Stand. (U.S.). 80(2); 291–311; 1976.

Figures in brackets indicate literature references at the end of this paper.

The PFORT Verifier, A. D, Hall and B. G. Ryder, Bell Laboratories, Murray Hill, N.J. Proceedings of the Computer Science and Statistics Eighth Annual Symposium on the Interface, University of California, Los Angeles, February 13–14, 1975.

⁴

Program Execution Profiles, G. Sande, World Bank, Washington, D.C. Proceedings of the Computer Science and Statistics Eighth Annual Symposium on the Interface, University of California, Los Angeles, February 13–14, 1975.

11. References

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[6].Gautschi W. Efficient Computation of the Complex Error Function. SIAM J. Numer. Anal. 7,1: 187–198; 1970. [Google Scholar]
[7].Hummer D. G. The Voigt Function: An Eight-Significant Figure Table and Generating Procedure. Mem. Roy. Astronom, Soc. 70: 1–31, 1965. [Google Scholar]
[8].Karpov K. A. Tablitsy Funktsii $w (z) = e^{- z^{2}} \int_{0}^{x} e^{x^{z}} d x v$ Kompleksnoi Oblasti. Moscow, U.S.S.R.: Izdat. Akad. Nauk S.S.S.R.; 1954. [Google Scholar]
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[11].Kreyszig E. and Todd J. On the Radius of Univalence of the Function $e x p z^{2} \int_{0}^{z} e x p (- t^{2}) d t$ . Pacific Journal of Mathematics 9, 1: 123–127; 1959. [Google Scholar]
[12].Lohmander B. and Rittsten S. Table of the Function $y = e^{- x^{2}} \int_{0}^{x} e^{t^{2}} d t$ . Kungl. Fysiogr. Sallsk. i Lund Forh. 28: 45–52; 1958. [Google Scholar]
[13].Martz C. W. Tables of the Complex Fresnel Integral NASA Report SP-3010; Washington; 1964. [Google Scholar]
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[R2] [2].Faddeeva V. N. and Terent’ev N. M. Tables of Values of the Function $w (z) = e^{- z^{2}} [1 + (2 i / \sqrt π) \int_{0}^{z} e^{t^{2}} d t]$ for Complex Arguments. New York; English Trans., Pergamon Press; 1961. [Google Scholar]

[R3] [3].Fettis H. E., Caslin J. C. and Cramer K. R Complex Zeros of the Error Function and of the Complementary Error Function. Math. Comp. 27,122:401–407; 1973. [Google Scholar]

[R4] [4].Finn G. D. and Mugglestone D. Tables of the Line Broadening Function H(a,v). Mon. Not. R. Astr. Soc. 129 : 221–235; 1965. [Google Scholar]

[R5] [5].Fried B. D. and Conte S. D. The Plasma Dispersion Function, The Hilbert Transform of the Gaussian. New York; Academic Press; 1961. [Google Scholar]

[R6] [6].Gautschi W. Efficient Computation of the Complex Error Function. SIAM J. Numer. Anal. 7,1: 187–198; 1970. [Google Scholar]

[R7] [7].Hummer D. G. The Voigt Function: An Eight-Significant Figure Table and Generating Procedure. Mem. Roy. Astronom, Soc. 70: 1–31, 1965. [Google Scholar]

[R8] [8].Karpov K. A. Tablitsy Funktsii $w (z) = e^{- z^{2}} \int_{0}^{x} e^{x^{z}} d x v$ Kompleksnoi Oblasti. Moscow, U.S.S.R.: Izdat. Akad. Nauk S.S.S.R.; 1954. [Google Scholar]

[R9] [9].Karpov K. A. Tablitsy Funktsii $F (z) = \int_{0}^{z} e^{x^{2}} d x v$ Kompleksnoi Oblasti. Moscow, U.S.S.R.: Izdat Akad. Nauk S.S.S.R.; 1958. [Google Scholar]

[R10] [10].Kreyszig E. and Todd J. The Radius of Univalence of the Error Function. Numerische Mathematik 1: 78–89; 1959. [Google Scholar]

[R11] [11].Kreyszig E. and Todd J. On the Radius of Univalence of the Function $e x p z^{2} \int_{0}^{z} e x p (- t^{2}) d t$ . Pacific Journal of Mathematics 9, 1: 123–127; 1959. [Google Scholar]

[R12] [12].Lohmander B. and Rittsten S. Table of the Function $y = e^{- x^{2}} \int_{0}^{x} e^{t^{2}} d t$ . Kungl. Fysiogr. Sallsk. i Lund Forh. 28: 45–52; 1958. [Google Scholar]

[R13] [13].Martz C. W. Tables of the Complex Fresnel Integral NASA Report SP-3010; Washington; 1964. [Google Scholar]

[R14] [14].Pearcey T. Table of the Fresnel Integral. London, England; Cambridge University Press; 1956. [Google Scholar]

[R15] [15].Rosser J. B. Theory and Application of $\int_{0}^{x} e^{- x^{2}} d x$ and $\int_{0}^{z} e^{- p^{2} y^{2}} d y$ $\int_{0}^{y} e^{- x^{2}} d x$ . Brooklyn, NY: Mapleton House; 1948. [Google Scholar]

[R16] [16].Tablitsy Integralov Frenelya. Moscow, U.S.S.R.: Izdat. Akad. Nauk S.S.S.R.; 1953. [Google Scholar]

[R17] [17].Van Wijngaarden A. and Scheen W. L., Table of Fresnel Integrals. Verh. Nederl. Akad. Wetensch. Afd. Natuurk. Sec. I,19,4: 1–26; 1949. [Google Scholar]

PERMALINK

Automatic Computing Methods for Special Functions. Part IV. Complex Error Function, Fresnel Integrals, and Other Related Functions

Irene A Stegun

Ruth Zucker

Abstract

1. Introduction