Mathematical Methods for Physics and Engineering - Matematica.NET

COMPLEX VARIABLES24.7 Find the real and imaginary parts of the functions (i) z 2 , (ii) e z , and (iii) cosh πz.By considering the values taken by these parts on the boundaries of the region0 ≤ x, y ≤ 1, determine the solution of Laplace’s equation in that region thatsatisfies the boundary conditionsφ(x, 0) = 0, φ(0,y)=0,φ(x, 1) = x, φ(1,y)=y +sinπy.24.8 Show that the transformation∫ z1w =dζ0 (ζ 3 − ζ)1/2transforms the upper half-plane into the interior of a square that has one cornerat the origin of the w-plane and sides of length L, whereL =∫ π/20cosec 1/2 θdθ.24.9 The fundamental theorem of algebra states that, for a complex polynomial p n (z)of degree n, the equation p n (z) = 0 has precisely n complex roots. By applyingLiouville’s theorem (see the end of section 24.10) to f(z) =1/p n (z), prove thatp n (z) = 0 has at least one complex root. Factor out that root to obtain p n−1 (z)and, by repeating the process, prove the above theorem.24.10 Show that, if a is a positive real constant, the function exp(iaz 2 ) is analytic and→ 0as|z| →∞for 0 < arg z ≤ π/4. By applying Cauchy’s theorem to a suitablecontour prove that24.11 The function∫ ∞0cos(ax 2 ) dx =√ π8a .f(z) =(1− z 2 ) 1/2of the complex variable z is defined to be real and positive on the real axis inthe range −1