1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 605 Consider the circles lz-al = Ri, lz-bl = R 2 and lz-cl =Ra such that lb-aJ < R ...

606 Chapter^8 for all z E S. The constant w is called a period off. It is easy to see if w is a period off, then kw (k = ±1, ±2 ...

Sequences, Series, and Special Functions 607 y Fig. 8.l;l we have w = e2n:iz = -2n:y+2n:xi so that Hence if (8.19-4) we obtain o ...

608 Chapter^8 where () ;:::: Arg w and k is any integer. Then 1 1 () 2 . logw = - 2 . lnlwl + - 2 +k 1i'Z 1i'Z 1i' and it foll ...

Sequences, Series, and Special Functions 609 which is absolutely convergent in the ring r 2 < lwl < ri and uniformly conve ...

610 Chapter 8 so that dw =^2 7l'Z 'd X w Also, Hence {8.19-8) becomes An= 11 f(x + ib)e-2 ... in(x+ib) dx {8.19-9) If the stri ...

Sequences, Series, and Special Functions From (8.19-9) it follows that and Also, an= 1 1 f(x + ib)[e-21l"in(x-Hb) + e21l"in(x+ib ...

612 Chapter^8 = i [1+2 f)-l)ne2'1rinzl n=l valid for e-^2 .,..iz < 1, or, Im z > 0. Also, we may write F( w) = -i (1 -! 1 ...

Sequences, Series, and Special Functions has a Fourier expansion +co F(z) = L Ane+2.,..inz n=-oo valid in the strip -b < Im z ...

614 Chapter^8 This integral clearly diverges for Re z :::; 0. Suppose that z lies in a compact set ]{ contained in the half-plan ...

Sequences, Series, and Special Functions 615 Corollary 8.23 The gamma function can be represented in the form oo ( l)n r(z) =I: ...

616 Chapter^8 with respect to z, and also with respect to (, whenever Re z > 0 and Re(> 0. Proof By means of the transform ...

Sequences, Series, and Special Functions 617 Proofs (1) I'(l) = ft' e-t dt = 1. (2) We have r(1/ 2 ) = J 000 e-tr^1!^2 dt, and l ...

618 Chapter^8 = (z + n -l)(z + n - 2)r(z + n - 2) = (z + n - l)(z + n - 2) · · · (z + l)zr(z) (8.20-11) By writing this formula ...

Sequences, Series, and Special Functions 619 for n > 0 yields lira r(z) = 00 z-+-n (8) Suppose for the moment that 0 < Rez ...

620 Chapter 8 Replacing z by z + 1, the result above can be written in the form of a Laplace transform, namely, .C {tz} = l 00 e ...

Sequences, Series, and Special Functions since I 1 ( :: ~ ) I = 4lx^2 ~ y^2 I and Ju-v = lx-yl Finally,· letting u = v + t^2 in ...

622 Chapter^8 We have roo r t n lo e-ttz-i dt - lo ( 1 --;i) tz-i dt =in [e-t - (1- ~) n] tz-i dt + 100 e-ttz-i dt Given e > ...

Sequences, Series, and Special Functions 623 The formula I z r( )^1 . n.n z = 1m n--+oo z(z + 1) .. · (z + n) (8.20-19) was know ...

624 Chapter 8 If in (8.20-20) we let n = 1, 2, 3, ... , n - 1 and add the resulting inequalities, we obtain [ ( ) 2 3 2 n n-1 l ...