## Transcribed Text

5 Homework 5
1. Show that 2 sinz · sin w = cos(z − w) − cos(z + w) for any z, w ∈ C.
2. Find power series expansion of f(z) = log(4 + 3z) at the point z = −1.
3. If α > 1, show: Q∞
n=1
1 −
z
nα
converges uniformly on compact subsets of C.
4. The famous Fiboniacci numbers is a sequence defined by a0 = a1 = 1, an = an−1+an−2
for n ≥ 2. Compare this sequence with the coefficients of the power series centered at
0 of the function 1
z
2+z−1
.
6 Homework 6
1. Find the image of the right half-plane Re(z) > 0 under the linear transformation
w = f(z) = i(1−z)
1+z
.
2. Find the linear transformation w = f(z) that maps the points z1 = −i, z2 = 0, z3 = i
onto w1 = −1, w2 = i and w3 = 1 respectively.
3. Find the fixed points of w =
4z+3
2z−1
(by fixed point, we mean the point z0 such that
f(z0) = z0).
4. Find a linear transformation w = f(z) such that it maps the disk ∆(2) onto the right
half-plane {w | Re(w) > 0} satisfying f(0) = 1 and arg f′
(0) = π
2
.
7 Homework 7
1. Show: f(z) = x
2 + iy2
satisfies the Cauchy-Riemann equation at the origin, but is not
holomorphic in any neighborhood of 0.
2. Show: f(z) = |z|
2 + z
2
is not a holomorphic function.
3. Let f(z) = x
2 + axy + by2 + i(cx2 + dxy + y
2
). What kind of values for the constants
a, b, c, d, should we take so that f is holomorphic on C?
4. If f(z) is a holomorphic function with u(x, y) = 2(x − 1)y, f(2) = −i. Find f(z) by
consideration of the Cuachy-Riemann equation.
5. If f(z) = u+iv is a holomorphic function on a domain D satisfying |f(z)| = constant,
show that f(z) is constant.

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.