Relation To The Gamma Function
The integrand is an even function,
Thus, after the change of variable, this turns into the Euler integral
where Γ is the gamma function. This shows why the factorial of a half-integer is a rational multiple of . More generally,
Read more about this topic: Gaussian Integral
Famous quotes containing the words relation to the, relation to, relation and/or function:
“Any relation to the land, the habit of tilling it, or mining it, or even hunting on it, generates the feeling of patriotism. He who keeps shop on it, or he who merely uses it as a support to his desk and ledger, or to his manufactory, values it less.”
—Ralph Waldo Emerson (1803–1882)
“Among the most valuable but least appreciated experiences parenthood can provide are the opportunities it offers for exploring, reliving, and resolving one’s own childhood problems in the context of one’s relation to one’s child.”
—Bruno Bettelheim (20th century)
“... a worker was seldom so much annoyed by what he got as by what he got in relation to his fellow workers.”
—Mary Barnett Gilson (1877–?)
“To make us feel small in the right way is a function of art; men can only make us feel small in the wrong way.”
—E.M. (Edward Morgan)