Digital Library of Mathematical Functions
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7 Error Functions, Dawson’s and Fresnel IntegralsProperties

§7.13 Zeros

Contents

§7.13(i) Zeros of erfz

erfz has a simple zero at z=0, and in the first quadrant of there is an infinite set of zeros zn=xn+yn, n=1,2,3,, arranged in order of increasing absolute value. The other zeros of erfz are -zn, z¯n, -z¯n.

Table 7.13.1 gives 10D values of the first five xn and yn. For graphical illustration see Figure 7.3.5.

Table 7.13.1: Zeros xn+yn of erfz.
n xn yn
1 1.45061 61632 1.88094 30002
2 2.24465 92738 2.61657 51407
3 2.83974 10469 3.17562 80996
4 3.33546 07354 3.64617 43764
5 3.76900 55670 4.06069 72339

As n

7.13.1 xn λ-14μλ-1+116(1-μ+12μ2)λ-3-,
yn λ+14μλ-1+116(1-μ+12μ2)λ-3+,

where

7.13.2 λ =(n-18)π,
μ =ln(λ2π).

§7.13(ii) Zeros of erfcz

In the sector 12π<phz<34π, erfcz has an infinite set of zeros zn=xn+yn, n=1,2,3,, arranged in order of increasing absolute value. The other zeros of erfcz are z¯n. The zeros of w(z) are zn and z¯n.

Table 7.13.2 gives 10D values of the first five xn and yn. For graphical illustration see Figure 7.3.6.

Table 7.13.2: Zeros xn+yn of erfcz.
n xn yn
1 -1.35481 01281 1.99146 68428
2 -2.17704 49061 2.69114 90243
3 -2.78438 76132 3.23533 08684
4 -3.28741 07894 3.69730 97025
5 -3.72594 87194 4.10610 72847

As n

7.13.3 xn -λ+14μλ-1-116(1-μ+12μ2)λ-3+,
yn λ+14μλ-1+116(1-μ+12μ2)λ-3+,

where

7.13.4 λ =(n-18)π,
μ =ln(2λ2π).

§7.13(iii) Zeros of the Fresnel Integrals

At z=0, C(z) has a simple zero and S(z) has a triple zero. In the first quadrant of C(z) has an infinite set of zeros zn=xn+yn, n=1,2,3,, arranged in order of increasing absolute value. Similarly for S(z). Let zn be a zero of one of the Fresnel integrals. Then -zn, z¯n, -z¯n, zn, -zn, z¯n, -z¯n are also zeros of the same integral.

Tables 7.13.3 and 7.13.4 give 10D values of the first five xn and yn of C(z) and S(z), respectively.

Table 7.13.3: Complex zeros xn+yn of C(z).
n xn yn
1 1.74366 74862 0.30573 50636
2 2.65145 95973 0.25290 39555
3 3.32035 93363 0.22395 34581
4 3.87573 44884 0.20474 74706
5 4.36106 35170 0.19066 97324

As n the xn and yn corresponding to the zeros of C(z) satisfy

7.13.5 xn λ+α(απ-4)8πλ3+,
yn α2λ+,

with

7.13.6 λ =4n-1,
α =(2/π)ln(πλ).
Table 7.13.4: Complex zeros xn+yn of S(z).
n xn yn
1 2.00925 70118 0.28854 78973
2 2.83347 72325 0.24428 52408
3 3.46753 30835 0.21849 26805
4 4.00257 82433 0.20085 10251
5 4.47418 92952 0.18768 85891

As n the xn and yn corresponding to the zeros of S(z) satisfy (7.13.5) with

7.13.7 λ =2n,
α =(2/π)ln(πλ).

§7.13(iv) Zeros of (z)

In consequence of (7.5.5) and (7.5.10), zeros of (z) are related to zeros of erfcz. Thus if zn is a zero of erfcz7.13(ii)), then (1+)zn/π is a zero of (z).

For an asymptotic expansion of the zeros of 0zexp(12πt2)t (=(0)-(z) =C(z)+S(z)) see Tuẑilin (1971).