Mathematical Methods for Physicists: A concise introduction - Site Map

FUNCTIONS OF A COMPLEX VARIABLE
6.24 Evaluate H C dz=…z a†n ; n ˆ 2; 3; 4; ... where z ˆ a is inside the simple
closed curve C.
6.25 If f …z† is analytic in a simply-connected region R, and a and z are any two
points in R, show that the integral
Z z
a
f …z†dz
is independent of the path in R joining a and z.
6.26 Let f …z† be continuous in a simply-connected region R and let a and z be
z
points in R. Prove that F…z† ˆR a f …z 0 †dz 0 is analytic in R, and F 0 …z† ˆf …z†.
6.27 Evaluate
I
(a)
IC
(b)
C
sin z 2 ‡ cos z 2
…z 1†…z 2† dz
e 2z
…z ‡ 1† 4 dz,
where C is the circle jzj ˆ1.
6.28 Evaluate
I
C
2 sin z 2
…z 1† 4 dz;
where C is any simple closed path not passing through 1.
6.29 Show that the complex sequence
z n ˆ 1
n n2 1
i
n
diverges.
6.30 Find the region of convergence of the series P 1
nˆ1 …z ‡ 2†n‡1 =…n ‡ 1† 3 4 n .
6.31 Find the Maclaurin series of f …z† ˆ1=…1 ‡ z 2 †.
6.32 Find the Taylor series of f …z† ˆsin z about z ˆ =4, and determine its circle
of convergence. (Hint: sin z ˆ sin‰a ‡…z a†Š:†
6.33 Find the Laurent series about the indicated singularity for each of the
following functions. Name the singularity in each case and give the region
of convergence of each series.
(a) …z 3† sin 1
z ‡ 2 ; z ˆ2;
z
(b)
…z ‡ 1†…z ‡ 2† ; z ˆ2;
1
(c)
z…z 3† 2 ; z ˆ 3:
6.34 Expand f …z† ˆ1=‰…z ‡ 1†…z ‡ 3†Š in a Laurent series valid for:
(a) 1< jzj < 3, (b) jzj > 3, (c) 0< jz ‡ 1j < 2.
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