Math 6720
HW6
Appl. Complex Var., Asymptc Mthds
NAME: .....................................................................
Alexander Balk due 2/24/2016
1.
Describe singularities (their locations, types, and orders).
(a) What are the singularities of f ( z ) =
(b) Consider the multi-valued function f (
[( z +1)
10 z
− 1] cos
2
( z +1)
3
) =
( z/ 2) z ( z − π )
3
√ z − 1 z − 1
.
.
To separate f ( z ) into single-valued branches, make a brunch cut along the negative real semi-axis. What isolated singularities do these branches have?
(c) Consider the multi-valued function f ( z ) = (1 + z )
1 /z
.
To separate f ( z ) into single-valued branches, make a brunch cut along the negative real semi-axis. What isolated singularities do these branches have?

2.
The singularities of Euler’s gamma-function Γ( z ) .
(a) What are the singularities of Γ( z )?
(b) What are the residues of this function (at all singularities)?

3. The function f ( z ) is analytic in the entire complex plane except at z = i/ 2, where it has a simple pole, and at z = 2, where it has a double pole. It is known that
I f ( z ) dz = 2 πi ,
I
| z | =1 f ( z ) dz = 0 ,
| z | =3
I
| z | =3 f ( z )( z − 2) dz = 0 , and f ( z ) is bounded at infinity (i.e.
∃ M > 0 , ∃ R > 0 : | z | > R ⇒ | f ( z ) | < M ).
Find f ( z ) (unique up to an arbitrary additive constant).
Suggestion: Find an examle f
0
( z ) of such function and use the Liouville theorem to show that f ( z ) − f
0
( z ) is a constant.

4. Prove that if all singularities of an analytic function f ( z ) in the extended complex plane
C are poles, then f ( z ) is a rational function (i.e. the ratio of two polynomials).
Suggestion: First show that f ( z ) can have only finite number of poles. Then for each pole at finite z
0 take the negative power part of the corresponding Laurent expansion; for z
0
= ∞ , take the positive power part; add all of them to get a rational function R ( z ). Finally, apply the Liouville theorem to show that f ( z ) − R ( z ) is a constant.

5. Evaluate the integrals
Z
∞
−∞
Z
∞
( x + b ) 2 + a 2 sin kx
−∞ cos kx
( x + b ) 2 + a 2 dx =
π a dx = −
π a e
− ka e
− ka cos bk , sin bk
(with parameters k > 0 , a > 0 , b real).

6. Integrate
(with parameters a ≥ 0 , b ≥ 0).
Z
∞
−∞ cos ax − cos bx x 2 dx = π ( b − a )

7. Integrate
Z
∞ sin αx
−∞ sinh πx dx = tanh
α
2
In which strip a < Re( α ) < b of complex parameter α does this formula hold?
Suggestion: Notice that sinh π ( z + i ) = − sinh πz and use a long rectangular contour with indentations. Replace sin αx by the exponential function (at the end you will take the imaginary part) and notice that when x is replaced by x + i the new integrand is multiplied by a constant factor, independent of x .
At first, assume real α ; then use analytic continuation to establish this formula for complex α .