Definite Integral to Infinity of Exponential of -(a x^2 plus b over x^2)
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Theorem
- $\ds \int_0^\infty \map \exp {-\paren {a x^2 + \frac b {x^2} } } \rd x = \frac 1 2 \sqrt {\frac \pi a} \map \exp {-2 \sqrt {a b} }$
where $a$ and $b$ are strictly positive real numbers.
Proof
| \(\ds \int_0^\infty \map \exp {-\paren {a x^2 + \frac b {x^2} } } \rd x\) | \(=\) | \(\ds \int_0^\infty \map \exp {-a \paren {x^2 + \frac b {a x^2} } } \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int_0^\infty \map \exp {-a \paren {\paren {x - \frac 1 x \sqrt {\frac b a} }^2 + 2 \sqrt {\frac b a} } } \rd x\) | Completing the Square | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \exp {-2 \sqrt {a b} } \int_0^\infty \map \exp {-a \paren {x - \frac 1 x \sqrt {\frac b a} }^2} \rd x\) | Exponential of Sum | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \map \exp {-2 \sqrt {a b} } \int_{-\infty}^\infty \map \exp {-a \paren {x - \frac 1 x \sqrt {\frac b a} }^2} \rd x\) | Definite Integral of Even Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \map \exp {-2 \sqrt {a b} } \int_{-\infty}^\infty \map \exp {-a u^2} \rd u\) | Glasser's Master Theorem | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \exp {-2 \sqrt {a b} } \int_0^\infty \map \exp {-a u^2} \rd u\) | Definite Integral of Even Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \sqrt {\frac \pi a} \map \exp {-2 \sqrt {a b} }\) | Definite Integral to Infinity of $\map \exp {-a x^2}$ |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definite Integrals involving Exponential Functions: $15.78$