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In: Physics

Given a function φ(z) with z = x+iy let    U(x, y) = ½ [φ(x+iy) +...

  1. Given a function φ(z) with z = x+iy let

   U(x, y) = ½ [φ(x+iy) + φ(x-iy)] and V(x, y) = i/2 [φ(x+iy) –φ(x-iy)]

A) For φ(z) = z2 find U and V and their induced vector fields E =▼U and

F =▼V also show that ▼2U = ▼2V = 0

B) Repeat for f(z) = z3

C) For f(z) = ln z we get U(x, y) = ½ ln (x2+y2) and V(x, y) = arctan (y/x) Find ▼U (electrostatic field) and ▼V (magnetic field) . Also, show that ▼2U = ▼2V = 0 for x2+y2 ǂ 0.

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