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

§7.5 Interrelations

7.5.1 F(z)=12π(-z2-w(z))=-12π-z2erf(z).
7.5.2 C(z)+S(z)=12(1+)-(z).
7.5.3 C(z)=12+f(z)sin(12πz2)-g(z)cos(12πz2),
7.5.4 S(z)=12-f(z)cos(12πz2)-g(z)sin(12πz2).
7.5.5 -12πz2(z)=g(z)+f(z).
7.5.6 ±12πz2(g(z)±f(z))=12(1±)-(C(z)±S(z)).

In (7.5.8)–(7.5.10)

7.5.7 ζ=12π(1)z,

and either all upper signs or all lower signs are taken throughout.

7.5.8 C(z)±S(z)=12(1±)erfζ.
7.5.9 C(z)±S(z)=12(1±)(1-±12πz2w(ζ)).
7.5.10 g(z)±f(z)=12(1±)ζ2erfcζ.
7.5.11 |(x)|2=f2(x)+g2(x),
x0,
7.5.12 |(x)|2=2+f2(-x)+g2(-x)-22cos(14π+12πx2)f(-x)-22cos(14π-12πx2)g(-x),
x0.

See Figure 7.3.4.

7.5.13 G(x)=πF(x)-12-x2Ei(x2),
x>0.

For Ei(x) see §6.2(i).