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/e794bed29bd4773d6d05b91608f202a07b8b814f)
- b.
![{\displaystyle 1^{7+i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09c61d8bb1ca9395fb8fe4d7f4cd3b221d7ac4df)
- c. Find all 4 roots of
![{\displaystyle {\sqrt {2}}+i{\sqrt {2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57ab8462d89af0c1edfe26a7cad00ebfbfb081a3)
- d
![{\displaystyle \left|e^{\log(2)+i\log({\sqrt {2}})}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0cea037892ab276158ca4b4f27173faef4f5b703)
2. State the Cauchy-Riemann equations for a complex valued function
. If you use symbols other then
and
indicate how they relate to these quantities.
3. State whether the give function is holomorphic on the set where it is defined.
- a.
![{\displaystyle \displaystyle 2z+2z^{2}-{\frac {1}{z}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c7f3cbda0286d355f2cdfc571b68149c705c352)
- b. Let
and let
.
- c.
where
satisfies ![{\displaystyle \displaystyle {\frac {\partial }{\partial {\bar {z}}}}g(z)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc3876ff13a842534926fab96d5bdc1fc88d43cf)
- d.
![{\displaystyle |z|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28fd4d7dcabf618d707c21bd08306c7b3aa8b68e)
4. Let
be a simple closed curve so that
lies in the interior of the region bounded by
.
- a. Suppose
and compute
![{\displaystyle \oint _{\gamma }z^{n}\,dz,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31a8545d5492e0f4162d2225fa95eea21303e692)
- simply writing the correct value without any explanation will not receive credit.
- b. We now consider the case corresponding to
. Please compute
![{\displaystyle \oint _{\gamma }z^{-1}\,dz,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e714d77d8b81f6af8c059d16bff513caa5bf9284)
- and explain your steps.
- c: Now suppose
and compute
![{\displaystyle \oint _{\gamma }z^{n}\,dz.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c39c046bfe51889a3d8d4ff01e354154571995ef)
5. Let
be given by
. Calculate
6. Let
find a function
so that
is holomorphic in the complex plane and
.
7.
- a. Using the limit characterization of the complex derivative show that
is not holomorphic.
- b. On the other hand show that if
.
- c. Do parts (a) and (b) contradict each other, explain why or why not.
8. State Cauchy's integral theorem, and intuitively what you need to know about the function, domain and contour.