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7 Error Functions, Dawson’s and Fresnel IntegralsProperties

§7.2 Definitions

Contents
  1. §7.2(i) Error Functions
  2. §7.2(ii) Dawson’s Integral
  3. §7.2(iii) Fresnel Integrals
  4. §7.2(iv) Auxiliary Functions
  5. §7.2(v) Goodwin–Staton Integral

§7.2(i) Error Functions

7.2.1 erfz=2π0zet2dt,
7.2.2 erfcz=2πzet2dt=1erfz,
7.2.3 w(z)=ez2(1+2iπ0zet2dt)=ez2erfc(iz).

erfz, erfcz, and w(z) are entire functions of z, as is F(z) in the next subsection.

Values at Infinity

7.2.4 limzerfz =1,
limzerfcz =0,
|phz|14πδ(<14π).

§7.2(ii) Dawson’s Integral

7.2.5 F(z)=ez20zet2dt.

§7.2(iii) Fresnel Integrals

7.2.6 (z) =ze12πit2dt,
7.2.7 C(z) =0zcos(12πt2)dt,
7.2.8 S(z) =0zsin(12πt2)dt,

(z), C(z), and S(z) are entire functions of z, as are f(z) and g(z) in the next subsection.

Values at Infinity

7.2.9 limxC(x) =12,
limxS(x) =12.

§7.2(iv) Auxiliary Functions

7.2.10 f(z)=(12S(z))cos(12πz2)(12C(z))sin(12πz2),
7.2.11 g(z)=(12C(z))cos(12πz2)+(12S(z))sin(12πz2).

§7.2(v) Goodwin–Staton Integral

7.2.12 G(z)=0et2t+zdt,
|phz|<π.