Construction of function holomorphic in $\mathbb{C}\setminus\{0, 1\}$ satisfying specific conditions

Question

I'm learning about complex analysis, specifically (Laurent) series and residues, and need help with the following problem:

Construct a function $f(z)$ holomorphic in $\mathbb{C}\setminus\{0, 1\}$ with a pole of order $2$ at $z=0$, an essential singularity at $z=1$, with $\operatorname{Res}\left(f, 0\right)=1$ and $\operatorname{Res}\left(f, 1\right)=0$.

Here's my work so far:

Since we want $f(z)$ to have a pole of order $2$ at $z=0$, the Laurent expansion of our desired function must be of the form

$$f(z)=\frac{a_{-2}}{z^2}+\frac{a_{-1}}{z}+\sum_{n=1}^{\infty}a_nz^n \quad \quad (1)$$

where $a_{-2}\neq0$.

Moreover, since we want our function to have an essential singularity at $z=1$, then $f(z)$ must also be of the form

$$f(z)=\sum_{n=1}^{\infty}b_{-n}\frac{1}{(z-1)^n}+\sum_{n=0}^{\infty}b_{n}(z-1)^n \quad \quad (2)$$

such that $\forall n \in \mathbb{N}$, $\exists N > n:b_{-N} \neq 0$, i.e. the principal part of the Laurent expansion of the function $f(z)$ contains an infinite number of non-zero terms.

Given the above results my guess is that $f(z)$ will be the sum of two function satisfying the conditions $(1)$ and $(2)$ (respectively), that is

$$f(z) = g(z) + h(z)$$

where $g(z), h(z)$ statisfies $(1), (2)$ respectively.

Is my thinking correct so far? How do I continue from here and what are the steps to solve this kind of problem?

Answer 1

Following the explanation in the answers given by Amontillado and zhw., here's my solution to the exercise:

The function $g(z)=1/z+1/z^2$ takes care of the requirement at $0$. For the requirement at $1$, we use the exponential function. Note that the function $e^{1/(z-1)}$ has an essential singularity at $1$.Now we need to work out the requirement for the residue. Since we want $a_{-1}=0$ we need to subtract $1/(z-1)$ from the Laurent expansion of $e^{1/(z-1)}$. Thus, $h(z)=e^{1/(z-1)}-1/(z-1)$ takes care of the second requirement. It follows that one example of a function $f(z)$ satisfying the given requirements is

\begin{align} f(z)&=g(z)+h(z)\\\\ &=1/z+1/z^2+e^{1/(z-1)}-1/(z-1). \end{align}

Can someone verify is my work if correct?

Answer 2

Your thinking seems correct. Keep in mind that $Res(f,0)=a_{-1}$ in your $(1)$ equation. Also you don't need to have infinite nonzero $a_n$ to have a pole or an essential singularity. For the second part try to manipulate the exponential function to give an essential singularity.

Answer 3

Yes, you are thinking along correct lines, but you seem to be going for the most general form of a solution, when the problem asks for just one example. To do this, try the simplest things you can think of: For example $1/z + 1/z^2$ takes care of the the requirement at $0.$ And $g(1/(z-1)^2)$ takes care of the requirement at $1$, where $g$ is any entire function that is not a polynomial (think about the power series of $g$). Why not try adding them together!