2, 24) Prove chat the Cauchy—Riemann equations take on the following form in polar coordinates: ??/?r = 1/r ??/??      and     1/r ??/?? = - ??/?r

2, 24) Prove chat the Cauchy—Riemann equations take on the following form in polar coordinates:

??/?_r = 1/r ??/?_?      and     1/r ?_?/?_? = - ?_?/?_r .

2.25. For each of the following functions u, find a function ? such that u + i? holomorphic in some region. Maximize that region.

(c) U(x, y) = 2x² + x + 1 – 2y²

2.18. Where are the following functions differentiable? Where are they holomorphic Determine their derivatives at points where they are differentiable.

(a) ?

 (z) = e^-x e^-iy

(c) f (z) = x² + i y²

(d) f(z) = e^x e^-iy

3.31. Let z = x + Iy and show that

(a) sin z = sin x cosh y + i cos x sinh y.

(b) cos z = cos x cosh y — i sin x sinh y.

3.32. Prove that the zeros of sin z are all real valued. Conclude that they are precisely the integer multiples of π.

3.33. Describe the images of the following sets under the exponential function exp(z):

(a) The line segment defined by z= iy, 0 < y <2π

(b) The line segment defined by z = 1 + iy, 0 < y < 2π

(c) The rectangle {z = x + iy EC: 0<X < 1, 0 <y < 2π}.

2.20. Prove: If f is holomorphic in the region G CC and always real valued, then f is constant in G. (Hint: Use the Cauchy—Riemann equations (2.3) to show that f¹ =0.