Question

Consider the following complex-variable function cosh a < T f(z) la! cosh πχ, a) Find all its singularities, state their nature and compute the residues b) Consider the rectangular contour y with vertices at tR and tRi. Evaluate 6 6 dz cosh πχ c) Using the previous result take the limit R-to prove that cosh ax (10] 2 cos (g Hint: remember that cosh(a + b) -cosh a cosh b + sinh a sinh b d) Why is the above result not applicable if lal > π? 3

Answer 1

SCansi bn he 1 az where houiny vertice cos hTZ e+e 옐1 %.et. , e e. a sinnple poe ing coithin conteut is Here, only Res ftz) dz residue .thehen Hence

21T e. TR TK TR TTR 2T1R Hence in-the limit R→ lequotionen reduced to Cosh moti) coshTx cosh T fa t e 2 Cosh TTX

substitute-a'implace. f a' 2 T an ete 2 cosh s d a 2 sec. cosh cosh ar Co.Shay litis a even function + χ even function of x The. above equatton gives 2 Cosha eer cosh a cosh r )