Definite Integral to Infinity of Exponential of -(a x^2 plus b x plus c)

Theorem

 * $\displaystyle \int_0^\infty \map \exp {-\paren {a x^2 + b x + c} } \rd x = \frac 1 2 \sqrt {\frac \pi a} \map \exp {\frac {b^2 - 4 a c} {4 a} } \map \erfc {\frac b {2 \sqrt a} }$

where $\erfc$ denotes the complementary error function.