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

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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
$a$, $b$ and $c$ are real numbers with $a > 0$.


Proof

\(\displaystyle \int_0^\infty \map \exp {-\paren {a x^2 + bx + c} } \rd x\) \(=\) \(\displaystyle \int_0^\infty \map \exp {-a \paren {x + \frac b {2 a} }^2 + \frac {b^2} {4 a} - c} \rd x\) Completing the Square
\(\displaystyle \) \(=\) \(\displaystyle \map \exp {\frac {b^2 - 4 a c} {4 a} } \int_0^\infty \map \exp {-a \paren {x + \frac b {2 a} }^2} \rd x\) Exponential of Sum
\(\displaystyle \) \(=\) \(\displaystyle \map \exp {\frac {b^2 - 4 a c} {4 a} } \int_0^\infty \map \exp {-\paren {\sqrt a x + \frac b {2 \sqrt a} }^2} \rd x\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {\sqrt a} \map \exp {\frac {b^2 - 4 a c} {4 a} } \int_{\frac b {2 \sqrt a} }^\infty \map \exp {-u^2} \rd u\) substituting $u = \sqrt a x + \dfrac b {2 \sqrt a}$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 2 \sqrt {\frac \pi a} \map \exp {\frac {b^2 - 4 a c} {4 a} } \paren {\frac 2 {\sqrt \pi} \int_{\frac b {2 \sqrt a} }^\infty \map \exp {-u^2} \rd u}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 2 \sqrt {\frac \pi a} \map \exp {\frac {b^2 - 4 a c} {4 a} } \map \erfc {\frac b {2 \sqrt a} }\) Definition of Complementary Error Function

$\blacksquare$


Sources