Analytic function from u = log√(x²+y²)

Definition

A function f(z) is analytic at a point if it is differentiable at that point and in some neighbourhood of it. It is analytic in a region if it is analytic at every point of the region.

Solution

u = log√(x² + y²) = ½ log(x² + y²).

u_x = x/(x² + y²)
u_y = y/(x² + y²)

(u is harmonic: u_xx + u_yy = (y² − x²)/(x² + y²)² + (x² − y²)/(x² + y²)² = 0.)

Milne-Thomson method. Since f′(z) = u_x − i u_y, and the result holds for all x, y, put x = z, y = 0:

f′(z) = u_x(z, 0) − i u_y(z, 0) = 1/z − i·0 = 1/z

Integrate: f(z) = log z + c, where c is a constant.

Find v

log z = ln r + iθ, so f(z) = ln√(x² + y²) + i tan⁻¹(y/x) + c.

u = log√(x² + y²)
v = tan⁻¹(y/x) + constant

Check with the C–R equations: v_y = x/(x² + y²) = u_x, and v_x = −y/(x² + y²) = −u_y.

Answer: f(z) = log z + c, with v = tan⁻¹(y/x) + c′.