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:
“We must get back into relation, vivid and nourishing relation to the cosmos and the universe. The way is through daily ritual, and is an affair of the individual and the household, a ritual of dawn and noon and sunset, the ritual of the kindling fire and pouring water, the ritual of the first breath, and the last.”
—D.H. (David Herbert)
“Only in a house where one has learnt to be lonely does one have this solicitude for things. One’s relation to them, the daily seeing or touching, begins to become love, and to lay one open to pain.”
—Elizabeth Bowen (1899–1973)
“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)
“No one, however powerful and successful, can function as an adult if his parents are not satisfied with him.”
—Frank Pittman (20th century)