Let \(U\subset \mathbb C\) be an open set. Define what it means for a function \(f:U\to \mathbb C\) to be complex differentiable at a point \(z_0\in U\).
State the Cauchy-Riemann equations.
Let \(f:U\to \mathbb C\) be the function defined by \[f(z)=f(x+iy)=x\cos(y)+\sinh(iy)\cosh(x).\] Use the Cauchy-Riemann equations to determine the points \(z_0\in\mathbb C\) where \(f\) is complex differentiable.
On what subset of \(\mathbb C\) is the function \(f(z)=\left( z+i\right)^4-3\) conformal? Justify your response.
Describe the geometric effects of \(f(z)\) on the tangent vectors of the curves passing through the point \(z=1-2i\).
Let \(\gamma:[0,3]\to \mathbb
C\) be the contour given by \[\gamma(t):=\begin{cases}
2t,& \text{if }0\leq t\leq 1,\\
4-2i+2\left( -1+i\right)t,& \text{if }1\leq t\leq 2,\\
2\left( 3-t\right)i,& \text{if }2\leq t\leq 3.
\end{cases}\] (a) Sketch \(\gamma(t)\) in \(\mathbb C\).
(b) Evaluate \(\int_{\gamma}\cos(z)dz\).
Prove that for each \(a\in \mathbb R\), \(a>0\), the series \[\sum_{n=1}^\infty n^{-z}\] converges uniformly on \(\left\lbrace z\in\mathbb C\;:\; \mathrm{Re}(z)>1+a\right\rbrace\) where \(n^{-z}\) is defined using the principal logarithm [You may use without proof that \(\sum_{n=1}^\infty n^{-b}\), \(b\in\mathbb R\), \(b>1\), converges.]
Does the series \(\sum_{n=1}^\infty n^{-z}\) defines a continuous function on \(\left\lbrace z\in\mathbb C\;:\; \mathrm{Re}(z)>1\right\rbrace\)? Justify your response.
Does the series \(\sum_{n=1}^\infty n^{-z}\) converge uniformly on \(\left\lbrace z\in\mathbb C\;:\; \mathrm{Re}(z)\geq 1\right\rbrace\)? Justify your response.
Consider the set \(U=\mathbb C\setminus \left\lbrace iy\;:\;y\in\mathbb R,\;y\leq 0\right\rbrace\).
Sketch the set \(U\) in \(\mathbb C\).
Is \(U\) an open set? Justify your response.
Find a biholomorphic map from \(U\) to the open unit disc \(\mathbb{D}=\left\lbrace z\in \mathbb C\;:\; {\left|z\right|}<1\right\rbrace\) and justify why this map is biholomorphic.