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7 Error Functions, Dawson’s and Fresnel IntegralsProperties

§7.2 Definitions

Contents

§7.2(i) Error Functions

7.2.1 erfz=2π0ze-t2dt,
7.2.2 erfcz=2πze-t2dt=1-erfz,
7.2.3 w(z)=e-z2(1+2iπ0zet2dt)=e-z2erfc(-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|14π-δ(<14π).

§7.2(ii) Dawson’s Integral

7.2.5 F(z)=e-z20zet2dt.

§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)=(12-S(z))cos(12πz2)-(12-C(z))sin(12πz2),
7.2.11 g(z)=(12-C(z))cos(12πz2)+(12-S(z))sin(12πz2).

§7.2(v) Goodwin–Staton Integral

7.2.12 G(z)=0e-t2t+zdt,
|phz|<π.