NETMaths
Part BCSIR NET December 2023cr-orthogonality

Cr orthogonality

Let be a real-differentiable function and define u(x, y) = Re f(x + iy), v(x, y) = Im f(x + iy). Let denote the gradient. Which one of the following is necessarily true?

  1. A.For , the level curves and are orthogonal wherever they intersect.
  2. B. at every point.
  3. C.If f is an entire function, then at every point.
  4. D.If at every point, then f is an entire function.

Solution

For entire f the Cauchy–Riemann equations give . Without holomorphy there is no such relation, and orthogonal gradients alone do not force CR (e.g. f = z̄ has .

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

Related counterexample: A harmonic function on any domain has a harmonic conjugate

More on this topic

From Analytic FunctionsCauchy–Riemann equations, harmonic functions

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