Determine a function with
- An essential singularity at \(z=1\)
- An essential singularity at \(z=0\)
- A pole of order 1 at \(z=1-i\)
- A pole of order 2 at \(z=1+i\)
- A removable singularity at \(z=7\)
Solution
Note that writing a single function for each singularity and taking a product might work, except that there may be unforeseen cancellation of zeros of one with poles of another, or some might become removable. A surefire way is to take a sum, e.g. \[ f(z) = e^{1\over z-1} + e^{1\over z} + {1\over z - (1-i)} + {1\over (z - (1+i) )^2 } + { z-7 \over \sin(z-7) } .\]