6220-2020-08

Let \(f_i:\mathbb {D}\to \mathbb {C}\) be a sequence of injective holomorphic functions so that \(f_i\to f_\infty \) uniformly on compact sets. If \(f_\infty \) is not injective, what can you say about it? Prove this.

Prove that if \(f:\{z:1<|z|<2\}\to \mathbb {C}\) is holomorphic then \[ f(z)=\sum _{j=-\infty }^{\infty } a_j z^j. \] What can you say about the power series \(\sum _{j=0}^{\infty }a_jz^j\) and \(\sum _{j=-\infty }^{0}a_jz^{-j}\)? Prove this.

Prove or disprove: An entire function \(f\) so that \(f(n)=0\) for all \(n\in \mathbb {Z}\) is identically 0.

Let \(f:\mathbb {H}\to \mathbb {D}\) be a holomorphic function so that \(f(i)=0\). What can you say about \(|f(2i)|\)? (Of course with proof.)

Show that if \(f:\{z:0<|z|<1\}\to \mathbb {C}\) is meromorphic and has poles at \(\frac {1}{2},\frac {1}{3},\frac {1}{4},\ldots \) then the range of \(f\) is dense in \(\mathbb {C}\).

State the Cauchy Riemann equations and prove that a function is holomorphic on an open set iff it satisfies the Cauchy Riemann equations.

Use the methods of complex analysis to compute \[ \int _{-\infty }^{\infty }\frac {dx}{(1+x^2)^3}. \]