Complex Analysis Midterm Exam
May 1, 20201. (a) (6 points) Find explicitly
1
√2+ i
√2
2020
.
(b) (6 points) Solve the equation in polar form:z4=−1+√ 3i.
(c) (6 points) Find all solutions ofez=1+i.
2. (a) (6 points) Find a power series expansion for 1
z aroundz=1+i.
(b) (8 points) Using the identity 1 1−z =
∞ n=0
∑
znfor|z|<1,find closed forms for the sums
∞ n=1
∑
nznand
∞ n=1
∑
n2zn.
3. (6 points) Show that there are no analytic functions f =u+i vwithu(x,y) =x2+y2.
4. (6 points) Show that there is no power series f(z) =
∞ n=0
∑
cnznsuch that
i. f(z) =1 forz= 1 2, 1
3, 1 4, . . . , ii. f0(0)>0.
5. (a) (8 points) Prove that a nonconstant entire function cannot satisfy the two equations i. f(z+1) = f(z)
ii. f(z+i) = f(z)
for allz∈C. [Hint: Show that a function satisfying both equalities would be bounded.]
(b) (6 points) Suppose f is (complex) analytic in the diskD={z| |z| ≤1}and suppose that i. |f(z)| ≤2 for|z|=1,Imz≥0,
ii. |f(z)| ≤3 for|z|=1,Imz≤0.
Show that|f(0)| ≤√
6. [Hint: Consider f(z)·f(−z).]
6. (a) (6 points) Find the maximum and minimum moduli ofz2−zin the diskD={z| |z| ≤1}.
(b) (8 points) Suppose f and g are both analytic in a compact domain D. Show that |f(z)|+|g(z)|
takes its maximum on the boundary. [Hint: Consider f(z)eiα+g(z)eiβ for appropriateα andβ.]
7. (a) (8 points) Suppose that f is entire and that|f(z)| ≥ |z|N for sufficiently largez.Show that f must be a polynomial of degree at leastN.
(b) (8 points) Find all entire functions f(z)onCsatisfying
|f(z)| ≤ |z|ex, z=x+iy∈C.
8. (12 points) Show that Z ∞
0
sinx
x dx=π 2.
