Define Cauchy-Riemann equations also discuss CRE in polar form & Cartesian form
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e., complex differentiable) in a domain. If a complex function f(z) = u(x, y) + iv(x, y) is holomorphic, where u and v are real-valued functions of x and y, then it must satisfy the Cauchy-Riemann equations: Show more…
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1. Using Cauchy-Riemann equations of Cartesian form find the derivative (i.e f'(z)) of f(z) = z^2. 2. Show that the function f(z) = re^{i heta} has a derivative everywhere in its domain using the Cauchy-Riemann equations of polar form. 3. Show that u(x, y) = xe^x cos y - ye^x sin y is harmonic. 4. Find the harmonic conjugate v(x, y) of u(x, y) where u(x, y) = xe^x cos y - ye^x sin y. 5. Using Cauchy-Riemann equations determine whether f(z) = z^3 is analytic or not.
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Using polar coordinates (Problem 46), find out whether the following functions satisfy the Cauchy-Riemann equations. $$ |z|^{1 / 2} e^{i \theta i 2} $$
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