Gaussian Integral - Relation To The Gamma Function

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:

    “Much poetry seems to be aware of its situation in time and of its relation to the metronome, the clock, and the calendar. ... The season or month is there to be felt; the day is there to be seized. Poems beginning “When” are much more numerous than those beginning “Where” of “If.” As the meter is running, the recurrent message tapped out by the passing of measured time is mortality.”
    —William Harmon (b. 1938)

    “You must realize that I was suffering from love and I knew him as intimately as I knew my own image in a mirror. In other words, I knew him only in relation to myself.”
    —Angela Carter (1940–1992)

    “You know there are no secrets in America. It’s quite different in England, where people think of a secret as a shared relation between two people.”
    —W.H. (Wystan Hugh)

    “If the children and youth of a nation are afforded opportunity to develop their capacities to the fullest, if they are given the knowledge to understand the world and the wisdom to change it, then the prospects for the future are bright. In contrast, a society which neglects its children, however well it may function in other respects, risks eventual disorganization and demise.”
    —Urie Bronfenbrenner (b. 1917)