Compute the real integral cosh() with | sinh(x +a)|2d a (0,7/2) (2) with the help of the residue theorem by closing contours. (a) Relate the integral to an integration over a meromorphic function on R by noticing | sinh(x + ia)2 = sinh(x + ia) sinh(x ia) for any r ER and the definition of the cosh(z) in terms of e, see Eq. (1). (b) Construct a closed counter-clockwise rectangular contour that reflects the anti-periodicity of sinh, see Eq. (1) and apply the residue theorem. (c) Relate the two horizontal lines and show that the vertical lines vanish in the limit of an infinitely long rectangle. (d) Eventually compute the integral (2) with the construction above. Compute the real integral cosh() with | sinh(x +a)|2d a (0,7/2) (2) with the help of the residue theorem by closing contours. (a) Relate the integral to an integration over a meromorphic function on R by noticing | sinh(x + ia)2 = sinh(x + ia) sinh(x ia) for any r ER and the definition of the cosh(z) in terms of e, see Eq. (1). (b) Construct a closed counter-clockwise rectangular contour that reflects the anti-periodicity of sinh, see Eq. (1) and apply the residue theorem. (c) Relate the two horizontal lines and show that the vertical lines vanish in the limit of an infinitely long rectangle. (d) Eventually compute the integral (2) with the construction above