Question

(a) Find all the analytic functions (f) for which, for all (z in mathbb{C}) ( ext{Im } z = ext{Re } f(z)). Please provide full explanation to the answer and justify all your steps. (b) Find all singularities of the function (f(z) = frac{e^{-frac{1}{z}}}{z^2 - frac{pi^2}{4}}) and determine the nature of each of these singularities (e.g. removable singularity, simple pole, double pole, essential singularity etc.). Justify all your steps.

          (a) Find all the analytic functions (f) for which, for all (z in mathbb{C})
(	ext{Im } z = 	ext{Re } f(z)).
Please provide full explanation to the answer and justify all your steps.
(b) Find all singularities of the function
(f(z) = frac{e^{-frac{1}{z}}}{z^2 - frac{pi^2}{4}})
and determine the nature of each of these singularities (e.g. removable singularity, simple pole, double pole, essential singularity etc.). Justify all your steps.

$(a) Find all the analytic functions (f) for which, for all (z in mathbbC) ( extIm z = extRe f(z)). Please provide full explanation to the answer and justify all your steps. (b) Find all singularities of the function (f(z) = frace^-frac1zz^2 - fracpi^24) and determine the nature of each of these singularities (e.g. removable singularity, simple pole, double pole, essential singularity etc.). Justify all your steps.$

Added by Christy M.

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