IGNOU PHE 14 2026 - Show that the function satisfies the Cauchy-Riemann and Lap

Question

Show that the function $\omega = \sin z$ satisfies the Cauchy-Riemann and Laplace equations.

29 Jan 2026

Answer :

Word Count : 349

Let ( \omega = \sin z ), where ( z = x + iy ) and ( \omega = u + iv ) with ( u(x,y) ) and ( v(x,y) ) real functions. Then [ \sin z = \sin(x + iy) = \sin x \cosh y + i \cos x \sinh y. ] So, [ u(x,y) = \sin x \cosh y, \quad v(x,y) = \cos x \sinh y. ] The Cauchy-Riemann equations are [ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}. ] Compute the partial derivatives: [ \frac{\partial u}{\partial x} = \frac{\partial}{\partial x}(\sin x \cosh y) = \cos x \cosh y, ] [ \frac{\partial v}{\partial y} = \frac{\partial}{\partial y}(\cos x \sinh y) = \cos x \cosh y. ]

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