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

§7.11 Relations to Other Functions

Incomplete Gamma Functions and Generalized Exponential Integral

For the notation see §§8.2(i) and 8.19(i).

7.11.1 erfz =1πγ(12,z2),
7.11.2 erfcz =1πΓ(12,z2),
7.11.3 erfcz =zπE12(z2).

Confluent Hypergeometric Functions

For the notation see §13.2(i).

7.11.4 erfz=2zπM(12,32,-z2)=2zπ-z2M(1,32,z2),
7.11.5 erfcz=1π-z2U(12,12,z2)=zπ-z2U(1,32,z2).
7.11.6 C(z)+S(z)=zM(12,32,12πz2)=zπz2/2M(1,32,-12πz2).

Generalized Hypergeometric Functions

For the notation see §§16.2(i) and 16.2(ii).

7.11.7 C(z) =zF21(14;54,12;-116π2z4),
7.11.8 S(z) =16πz3F21(34;74,32;-116π2z4).