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If $v = e^x siny$, Prove that v is a harmonic function. Also find the corresponding harmonic conjugate function and analytic function.
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written 7.8 years ago by |
$$v = e^x siny$$ $$\frac{\partial v}{\partial x} = e^x siny \hspace{1cm} \frac{\partial v}{\partial y} = e^x cosy$$ $$\frac{\partial^2 v}{\partial x^2} = e^x siny \hspace{1cm} \frac{\partial^2 v}{\partial y^2} = -sinye^x$$ $$\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = e^x siny + (-e^x siny) = e^x siny - e^x siny$$ $$\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = 0$$
$\therefore$ v is a harmonic function.
$u_{x} = v_{y} \\ \therefore u_{x} = e^x cosy \\ u = \int uxdr = e^x cosy$
$f(z) = u + iv = e^x csy + i(e^x siny) = e^z + c$ (x = z & y = 0)
$f(z) = e^{(x + iy)} = e^x * e^{iy} = e^x (cosy + isiny) = e^x cosy + ie^x siny$
$\therefore v = e^x siny$ is harmonic conjugate.
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