Theorem: Laplace Transform of Gauss Error Function

:$\laptrans {\map \erf t} = \dfrac 1 s \map \exp {\dfrac {s^2} 4} \map \erfc {\dfrac s 2}$
where:
:$\laptrans f$ denotes the Laplace transform of the function $f$
:$\erf$ denotes the Gauss error function
:$\erfc$ denotes the complementary error function
:$\exp$ denotes the exponential function.

Proof:
{{begin-eqn}}
{{eqn | l = \laptrans {\map \erf {\sqrt t} }
      | r = \laptrans {\frac 2 {\sqrt \pi} \int_0^{\sqrt t} \map \exp {-u^2} \rd u}
      | c = {{Defof|Gauss Error Function}}
}}
{{eqn | r = \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} }
      | c = {{Defof|Exponential Function/Real|Real Exponential Function|subdef = Power Series Expansion}}
}}
{{eqn | r = \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!} } }
      | c = Primitive of Power
}}
{{eqn | r = \frac 2 {\sqrt \pi} \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1} n!} \laptrans {t^{n + \frac 1 2} }
      | c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \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} } 
      | c = Laplace Transform of Real Power
}}
{{eqn | r = \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}
      | c = Gamma Difference Equation
}}
{{eqn | r = \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} 
      | c = Gamma Function Extends Factorial
}}
{{end-eqn}}
We have: 
{{begin-eqn}}
{{eqn | l = \map \Gamma {n + \frac 1 2}
      | r = \frac \pi {\map \sin {\pi \paren {\frac 1 2 - n} } \map \Gamma {\frac 1 2 - n} }
      | c = Euler's Reflection Formula
}}
{{eqn | r = \frac \pi {\map \cos {-n \pi} \map \Gamma {\frac 1 2 - n} }
      | c = Sine of Complement equals Cosine
}}
{{eqn | r = \frac {\paren {-1}^n \pi} {\map \Gamma {\frac 1 2 - n} }
      | c = Cosine of Integer Multiple of Pi
}}
{{end-eqn}}
So: 
{{begin-eqn}}
{{eqn | l = \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} 
      | r = \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}
      | c = 
}}
{{eqn | r = \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}
      | c = 
}}
{{eqn | r = \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}
      | c = Gamma Function of One Half
}}
{{eqn | r = \dfrac 1 {s^{3/2} } \sum_{n \mathop = 0}^\infty \binom {-\frac 1 2} n \frac 1 {s^n}
      | c = {{Defof|Binomial Coefficient/Complex Numbers|Binomial Coefficient}}
}}
{{eqn | r = \dfrac 1 {s^{3/2} } \paren {1 + \dfrac 1 s}^{-1/2}
      | c = General Binomial Theorem
}}
{{eqn | r = \dfrac 1 {s \sqrt {s + 1} }
      | c = simplification
}}
{{end-eqn}}
{{qed}}