Sequences, Series, and Special Functions 617Proofs (1) I'(l) = ft' e-t dt = 1.
(2) We have r(1/ 2 ) = J 000 e-tr^1!^2 dt, and le~ting t = x^2 we obtaintr(1/2) = 2 fo00
e-"'2
dx = y'7r(3) By assuming that Rez > 0, integration by parts gives
I'(z + 1) = 1
00
e-ttz dt = -tze-t) ~ +z 100e-ttz-l dt
= zI'(z) (8;20-9)Formula (8.20-9) is also valid for any z other than 0, -1, -2, ... since the
function g(z) = I'(z + 1) - zI'(z) is analytic in D = C - {O, -1, -2, ... }
and g( z) = 0 for Re z > 0. Hence, by the identity principle for analyticfunctions (Theorem 8.24), g(z) = 0 for all z ED, or I'(z + 1) = zI'(z) for
all z except z = 0, -1, -2, ....
Formula (8.20-9) is called the functional equation of the r-function. It
does not characterize the r-function unless some additional conditions are
imposed. We discuss this matter in Selected Topics, Section 4.4. Thefunctional equation plays an important role in calculations with the r-
function. It tells us that once the values of I'(z) are known in some vertical
strip n - 1 < Re z ::::; n (n = 1, 2, ... ), the values of the function in some
other strip are easily evaluated.Example r(%) =^1 / 2 r(1/ 2 ) =^1 / 2 y'7r.
In C.R.C Standard Mathematical Tables the values of the real function
I'( x) are tabulated, with five decimals, for the interval 1 :5 x :5 2. This is
the interval that is used for best linear interpolation. In Jahnke-Emde a
table of r(x + 1) is given for the interval 0 < x ::::; 3.99.
By writing the functional equation in the formr(z) = r(z + 1)
z(8.20-10)the values of r( z) in the strip -1 < Re z < 0, z f:. 0, can be determined from
those in 0 <Re z ::::; 1. Then the values of I'(z) in the strip -2 < Re z ::::; -1,
z f:. -1, from those in -1 < Rez ::::; 0, z f:. 0, and so on.
Example r(-^1 / 2 ) = r(^1 / 2 )/(-^1 / 2 ) = -2.fi.
( 4) By applying (8.20-9) n times we find thatI'(z + n) = (z + n - l)I'(z + n - l)
tsee D. V. Widder [42), p. 371. This result was used in Section 7.15.