APPM 4360/5360 Introduction to Complex Variables and Applications
HOMEWORK #6
Assigned: Wednesday March 16, 2016
DUE: At class Wednesday April 6, 2016
XC: Extra Credit
* Solve: Discuss all singularities of the following functions; including the type of singularity: pole–include order, essential, branch point, cluster ..., that each of these functions
have in the finite z−plane. If the functions have a Laurent series around these singularities
find the first two nonzero terms
z 1/3 +1
z+1
a) sech(z + 2) b) cot(1/z) c)
* Solve: Consider
f (z) =
∞
X
d) ecot z
n
z3
n=0
Find a functional equation that f (z) satisfies; discuss where f (z) is analytic and its
singularities
* Solve: Evaluate the integral
1
2πi
H
f (z)dz where C is a unit circle centered at
C
the origin and f (z) is given below
a)
z+1
z 3 +a3 ,
0 < a < 1, b) sin(1/z)
c)
log(z+a)
z+1/a ,
a > 1, principal branch
* Solve: 4.1.2 a, c
* Solve Determine the type of singular point each of the following functions have at
infinity
a)
zN
z M +1
, N > M positive integers,
b) log(z 4 + 1) c) cos z
* Solve: 4.1.4
* Solve: 4.1.7
1
* Solve: Evaluate the following real integrals
a)
R∞
x2
−∞ (x2 +α2 )(x2 +β 2 ) dx,
α > 0, β > 0 b)
R∞
0
x2 +α2
(x2 +1)2 dx, α
>0
* Solve: 4.2.2 a, c, d
(XC) *Solve 4.1.5a,b and derive a formula for Jn (w); Jn (w) is called the Bessel function
order n of the first kind.
2