1550251515-Classical_Complex_Analysis__Gonzalez_

(jair2018) #1

Integration


2+i
(a) 1 (t^2 + 1 + it)dt

(b) 1i"'(et + i cost) dt



  1. Evaluate.


(a) J lzl dz, where 1: z = t +it, 0 :::; t :::; 1


'Y
(b) j (z + 3z) dz, where 1: z = t^2 - it, 0 :::; t :::; 2
'Y
(c) J xdz, where 1: z = a+reit, 0:::; t:::; 211"
'Y
(d) i lz + llldzl, where 1: z = eit, 0 :::; t:::; 211"

439

(e) J lzlzdz, where1: z =t, -1:::; t:::; 1, and1 1 : z = eit, 0:::; t:::; ?r


'Y+'Y1
(f) i z^2 dx, where 1: z = 2eit, 0 :::; t :::; %1!"

(g) J sinhzdz, where 1: z = 1 +it, 0:::; t:::; 1


'Y


  1. Show that


I


dz 3 ·
(a) z 2 +
1
:::;
16

?r, where 1: z = 3e'\ 0:::; t:::; 1j 2 ?r

'Y

(b) J ( e^2 z + 2z) dz :::; 108, where C is the boundary of the triangle


c

with vertices at 0, -3, and 4i


  1. Show that the equality sign holds in Darboux's inequality only if
    lf(z)I = M everywhere on I·

  2. Let P(z) be a polynomial and 1:


(a) J P(z)dz = 0


'Y
(c) j P(z)dz = -2?rir^2 P'(a)
'Y


  1. Show that:


z =a+ reit, 0:::; t:::; 211". Show that:


(b) j P(z) dz = o
'Y

(a) J ezz dz :::; e?r, where 1: z = ei\ 0 :::; t :::; 7r


'Y
Free download pdf