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Complex Function ( Limit, Analytic Function)

An analytic f...

Question

An analytic function $f (z)$ of complex variable $z = x + i y$ may be written as $f (z) = u (x, y) + i v (x, y)$ . Then $u (x, y)$ and $v (x, y)$ must satisfy.

\frac{\partial u}{\partial x} = - \frac{\partial v}{\partial y}

and

\frac{\partial u}{\partial y} = \frac{\partial v}{\partial x}

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\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}

and

\frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}

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\frac{\partial u}{\partial x} = - \frac{\partial v}{\partial y}

and

\frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}

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\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}

and

\frac{\partial u}{\partial y} = \frac{\partial v}{\partial x}

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Solution

The correct option is B $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}$
$f (z) = u (x, y) + i v (x, y)$
$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ .... (i)
$\frac{\partial u}{\partial y} = - \frac{\partial y}{\partial x}$

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An analytic function f(z) of complex variable z=x+iy may be written as f(z)=u(x,y)+iv(x,y). Then u(x,y) and v(x,y) must satisfy.

The correct option is B ∂u∂x=∂v∂y and ∂u∂y=−∂v∂xf(z)=u(x,y)+iv(x,y) ∂u∂x=∂v∂y .... (i) ∂u∂y=−∂y∂x

An analytic function $f (z)$ of complex variable $z = x + i y$ may be written as $f (z) = u (x, y) + i v (x, y)$ . Then $u (x, y)$ and $v (x, y)$ must satisfy.