Solve the improper integral

(See attached file for full problem description) I tried solve this problem by following Cauchy's Residue Theorem. However, the answer is always wrong.

Evaluating the improper intergrals using residues theorem.

Evaluating the improper intergrals using residues theorem.

Is this function analytic

(See attached file for full problem description) Is this function analytic? Why? I can't find the value to show it.

Application of Residue theorem

Application of Residue theorem. See attached file for full problem description.

Locus of points that satisfy eqn eg circle, hyperbola, ect

In the following problems determine the set of Z satisfying the given equations 1. |Z|=|Z-i| 2. |Z|^2 + Im(z)= 16 3. |Z-i|+|Z| = 9 Where: Z = x+yi __ Z = x-yi