Laplace Transform of Error Function of Root/Proof 1
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Theorem
- $\laptrans {\map \erf {\sqrt t} } = \dfrac 1 {s \sqrt {s + 1} }$
where:
- $\laptrans f$ denotes the Laplace transform of the function $f$
- $\erf$ denotes the error function
Proof
\(\ds \laptrans {\map \erf {\sqrt t} }\) | \(=\) | \(\ds \laptrans {\frac 2 {\sqrt \pi} \int_0^{\sqrt t} \map \exp {-u^2} \rd u}\) | Definition of Error Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \laptrans {\frac 2 {\sqrt \pi} \int_0^{\sqrt t} \paren {\sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {u^{2 n} } {n!} \rd u} }\) | Definition of Real Exponential Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \laptrans {\frac 2 {\sqrt \pi} \paren {\sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {\sqrt t}^{2 n + 1} } {\paren {2 n + 1} n!} } }\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 {\sqrt \pi} \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1} n!} \laptrans {t^{n + \frac 1 2} }\) | Linear Combination of Laplace Transforms | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 {\sqrt \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\map \Gamma {n + \frac 3 2} } {\paren {2 n + 1} n! s^{n + \frac 3 2} }\) | Laplace Transform of Real Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 {s^{3/2} \sqrt \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {n + \frac 1 2} \map \Gamma {n + \frac 1 2} } {\paren {2 n + 1} n! s^n}\) | Gamma Difference Equation | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {s^{3/2} \sqrt \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\map \Gamma {n + \frac 1 2} } {\map \Gamma {n + 1} s^n}\) | Gamma Function Extends Factorial |
We have:
\(\ds \map \Gamma {n + \frac 1 2}\) | \(=\) | \(\ds \frac \pi {\map \sin {\pi \paren {\frac 1 2 - n} } \map \Gamma {\frac 1 2 - n} }\) | Euler's Reflection Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac \pi {\map \cos {-n \pi} \map \Gamma {\frac 1 2 - n} }\) | Sine of Complement equals Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {-1}^n \pi} {\map \Gamma {\frac 1 2 - n} }\) | Cosine of Integer Multiple of Pi |
So:
\(\ds \frac 1 {s^{3/2} \sqrt \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\map \Gamma {n + \frac 1 2} } {\map \Gamma {n + 1} s^n}\) | \(=\) | \(\ds \frac 1 {s^{3/2} \sqrt \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {-1}^n \pi} {\map \Gamma {n + 1} \map \Gamma {\frac 1 2 - n} s^n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {s^{3/2} } \sum_{n \mathop = 0}^\infty \frac {\sqrt \pi} {\map \Gamma {n + 1} \map \Gamma {\frac 1 2 - n} s^n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {s^{3/2} } \sum_{n \mathop = 0}^\infty \frac {\map \Gamma {-\frac 1 2 + 1} } {\map \Gamma {n + 1} \map \Gamma {-\frac 1 2 - n + 1} s^n}\) | Gamma Function of One Half | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {s^{3/2} } \sum_{n \mathop = 0}^\infty \binom {-\frac 1 2} n \frac 1 {s^n}\) | Definition of Binomial Coefficient | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {s^{3/2} } \paren {1 + \dfrac 1 s}^{-1/2}\) | General Binomial Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {s \sqrt {s + 1} }\) | simplification |
$\blacksquare$
Sources
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: The Error Function: $39$