1550251515-Classical_Complex_Analysis__Gonzalez_

494

7. The Laguerre polynomial Ln(z) is given by

Ln(z) = ez Dn(zne-z)

Apply (7.21-2) to show that

I 1 (n -((-z)

Ln(z) = 2n.. (( e )n+l d(

7rZ c+ - Z

C being any simple closed contour around z.

8. If f is analytic on the open unit disk D = {z: lzl < 1}, and

1

lf(z)I S 1 _ lzl

for z E D, show that

IJCn>(o)I S (n + 1)!(1 + .!_ t < e(n + 1)!

n

Chapter 7

9. Let -y: z = z( s ), 0 S s S L, be any oriented rectifiable simple arc
   (not necessarily closed), parametrized in terms of the arc length s, and
   suppose that z'(so) f:. 0 exists at z 0 = z(s 0 ). In addition, suppose that
   f(z) satisfies for all z E 'Y the Holder condition ,

lf(z) - f(zo)I S Alz - zol"'

where A > 0 and a > 0 are real constants. If we let

F(z) = ~ J J(()d(

27rz ( - z

(zE1*)

-y+

then prove the formulas

F_(zo) = (PV) ~ J f(() d( + ~ f(zo)

27rz ( - z 0 2

'Y

F+(zo) = (PV)~ J J(()d( - ~f(zo)

27rz ( - zo 2

'Y

so that

F_(zo) - F+(zo) = J(zo)

1 J f(()d(

F_(zo) + F+(z 0 ) = (PV)---:- ;-

7rZ ':, - Zo

(1)

(2)

where F_(z 0 ) denotes the limit of F(z) as z ~ z 0 along a nontangential

path contained in the left-hand portion cut out by 1* on Ns(zo) (c5 > 0