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7 Error Functions, Dawson’s and Fresnel IntegralsProperties

§7.13 Zeros

Contents
  1. §7.13(i) Zeros of erfz
  2. §7.13(ii) Zeros of erfcz
  3. §7.13(iii) Zeros of the Fresnel Integrals
  4. §7.13(iv) Zeros of (z)

§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+iyn, 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+iyn 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 λ14μλ1+116(1μ+12μ2)λ3,
yn λ+14μλ1+116(1μ+12μ2)λ3+,

where

7.13.2 λ =(n18)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+iyn, 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 izn and iz¯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+iyn 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+14μλ1116(1μ+12μ2)λ3+,
yn λ+14μλ1+116(1μ+12μ2)λ3+,

where

7.13.4 λ =(n18)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+iyn, 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, izn, izn, iz¯n, iz¯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+iyn 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(απ4)8πλ3+,
yn α2λ+,

with

7.13.6 λ =4n1,
α =(2/π)ln(πλ).
Table 7.13.4: Complex zeros xn+iyn 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+i)zn/π is a zero of (z).

For an asymptotic expansion of the zeros of 0zexp(12πit2)dt (=(0)(z) =C(z)+iS(z)) see Tuẑilin (1971).