Cauchy-Riemann equations
General · Mathematics
Study notes
Q: Check C-R for f(z) = z² and f(z) = z̄. z²: u = x²-y², v = 2xy. uₓ = 2x, v_y = 2x ✓. u_y = -2y, -vₓ = -2y ✓. ANALYTIC! z̄: u = x, v = -y. uₓ = 1, v_y = -1 ✗ FAILS. Not analytic anywhere!
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General · Mathematics
Q: Check C-R for f(z) = z² and f(z) = z̄. z²: u = x²-y², v = 2xy. uₓ = 2x, v_y = 2x ✓. u_y = -2y, -vₓ = -2y ✓. ANALYTIC! z̄: u = x, v = -y. uₓ = 1, v_y = -1 ✗ FAILS. Not analytic anywhere!