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

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