This exam has a total of 100 points. You have 50 minutes. Partial credit will be awarded so showing your work can only help your grade
Question 1: Restrict
, and take the corresponding branch of the logarithm:
- (a)
![{\displaystyle \log(1+i{\sqrt {3}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/440c90359fc91232e64050c0e9c673610314901c)
- (b)
![{\displaystyle (1+i)^{1+i}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b40e16f53dd86c6e84a0f42fb3ab23b2a783e16)
- (c)
![{\displaystyle \sin(i\pi )\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b983d09f433fb99e0ec1046d2646a32c051e0d28)
- (d)
![{\displaystyle \left|e^{i\pi ^{2}}\right|\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e94b6d26cd3e5bce849f7040b23c7639fc8623d)
Question 2: Compute the following line integrals:
- (a) Let
for
. Compute the line integral
![{\displaystyle \oint _{\gamma }{\frac {1}{z^{3}}}\,dz}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c4f463a81af9bd3e38ed955b15c3d293fdde19b)
- (b) Let
for
. Compute the line integral
![{\displaystyle \oint _{\gamma }{\bar {z}}z^{2}\,dz}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c99073ab2f8280d11e744a9033b97fd22fdf120b)
- (c) Let
for
. Compute the line integral
![{\displaystyle \oint _{\gamma }e^{z}\cos(z)\,dz}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51fc3ab8434e47c3823396e34bb2ab1849fb3b96)
Question 3: Let
, verify that
is harmonic
and find a function
so that
and
is a
holomorphic function.
Question 4: Explain why there is no complex number
so that
.
Question 5: We often use the formulas from ordinary calculus to compute complex derivatives. This problem is part of the justification. Show that if
is holomorphic then
.
Comment: This problem shows that if
and
is a function in the complex plane, and
and
, then we can use this problem to show that
. We will
see later that if two holomorphic functions agree on a line then they agree everywhere. So it would have to be the case that
. (For example, take
and
then
, so it must be that
.)
Question 6: Decide whether or not the following functions are holomorphic where they are defined.
- (a)
![{\displaystyle f(z)={\frac {ze^{z}}{z-1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3ac8a2160a4ae711f905e9a6b214d14ea93b3fd)
- (b)
![{\displaystyle f(z)=e^{|z|^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe7ada36b0890a0ececaf27e6b92d9e458bdbaca)
- (c) Let
and let ![{\displaystyle f(x+iy)=x^{3}+xy^{2}+i(x^{2}y+y^{3})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7de6bc7f41e05c962553238950d9a6c6c5411741)
- (d) Let
and let ![{\displaystyle f(z)=re^{-i\theta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92cb87701c598b39e00d8d51bebc84e82c4c9483)
- (e) Let
and let ![{\displaystyle f(z)=e^{ix}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ee222659dba1b55f992b45efd3dc0ecf39e3064)
Question 7: State 4 ways to test if a function
is holomorphic.