2. (Cauchy-Riemann's equations, analyticity and harmonic functions) (a) Define the symbols ðf /dž and ôf /dz by af_1 (af 1af dz 2 dx i dy, af _1(af ¸ 1af dz 2 (dr" i dy, as suggested by the relations...

Extracted text: 2. (Cauchy-Riemann's equations, analyticity and harmonic functions) (a) Define the symbols ðf /dž and ôf /dz by af_1 (af 1af dz 2 dx i dy, af _1(af ¸ 1af dz 2 (dr" i dy, as suggested by the relations z = }(2+ 2), y = ±(z – =) and the chain rule. Show that the Cauchy-Riemann equations are equivalent to df/ðz = 0. Also, show that if f is analytic, then f' = df/dz. (b) Determine all functions f = u + iv that are analytic in the whole plane and has the property that the real part u is a function of only y = Im z. The answer should be given as an expression in the variable z = x + iy.