Consider the following complex-variable function: f(x) = cosh az cosh Tz' |a| < π. a) Find all its singularities, state

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Consider the following complex-variable function: f(x) = cosh az cosh Tz' |a| < π. a) Find all its singularities, state their nature and compute the residues. b) Consider the rectangular contour y with vertices at ±R and ±R + i. Evaluate cosh az cosh TZ -dz. c) Using the previous result take the limit R→ ∞ to prove that 1 T cosh ar cosh Tx -dx = 2 cos () Hint: remember that cosh (a + b) = cosh a cosh b+ sinh a sinh b d) Why is the above result not applicable if |a| > ? o [6] [6] [10] [3]