(a) Use the definitions of the trigonometric and hyperbolic functions in terms o

Question

(a) Use the definitions of the trigonometric and hyperbolic functions in terms of exponentials to express sinh (z + i pi/2) and cosh (z + i pi) in terms of sinh (z) and/or cosh (z). Also show that if a Element R and beta Element R then sinh (|a|) lessthanorequalto | cosh (alpha + i beta)| lessthanorequalto cosh (alpha) and sinh (|alpha|) lessthanorequalto |sinh (alpha + i beta)| lessthanorequalto cosh (alpha). (b) Use a rectangular contour argument to evaluate J (rho) = integral^infinity _- infinity e^ipx dx/cos h (x), rho Element R. (c) Use a rectangular contour argument to relate the integral I = integral^infinity _0 sin (x) dx/sinh (x) to an integral for which the denominator of the integrand is cosh (x).

Explanation / Answer

Hyperbolic functions are analogs of the ordinary trignometric or circular functions...Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the equilateral hyperbola.

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