Principles of Mathematics in Operations Research

13.7 Gamma Function

Definition 13.6.3 A trigonometric series is a series of the form

oo

f(x) = Y,

c

"

einx

>

xeR

-

— oo

If f is an integrable function on [—IT, IT], the numbers cm are called the Fourier

coefficients of f, and the series formed with these coefficients is called the

Fourier series of f.

13.7 Gamma Function

Definition 13.7.1 For 0 < x < oo,

/»oo

r(x)= / tx-le-ldt.

Jo

is known as the gamma function.

Proposition 13.7.2 Let F(x) be defined above.

(a) r(x + 1) = xr{x), 0 < x < oo.

(b) r(n + 1) = n\, n <= N. T(l) = 1.

(c) log-T is convex on (0,oo).

Proposition 13.7.3 If f is a positive function on (0, oo) such that

(a) f(x+l) = xf(x),

(b) /(I) = 1,

(c) log/ is convex.

then f{x) = r{x).

Proposition 13.7.4 If x,y G R+,

[\^{i-t)y^dt =

r

}f

r

^.

Jo r(x + y)

This integral is so-called beta function (3(x,y).

Remark 13.7.5 Let t = sin9, then

2 f~

2

(sing)

2

-

1

(cosefy^de =

r

};f

)r{y

}.

Jo ' r(x + y)

The special case x = y = | gives