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MATHMAA

# Proof of Cauchy - Riemann equations

Theorem :

The necessary conditions that a function  f(z)=U(x, y) + i V(x, y) be analytic in a domain D is that U and V satisfy the Cauchy- Riemann equations i.e.$$\frac{\partial U}{\partial x}=\frac{\partial V}{\partial y};\frac{\partial U}{\partial y}=-\frac{\partial V}{\partial x}$$.

Proof :

Let f(z) is analytic in D, then f'(z) exists uniquely at every point of D . Therefore for all z $$\epsilon$$ D

$${f}'(z)=\lim_{\Delta z\rightarrow 0} \frac{f(x+\Delta)-f(z)}{\Delta z}$$

exists, and is unique as $$\Delta z\rightarrow 0$$ along any path we choose.

Now we consider the following two cases:

Case I. Suppose first that in

$$\frac{f(z+\Delta z)-f(z)}{\Delta z}$$

$$\Delta z$$ approaches zero along the real axis or x-axis then $$\Delta z = \Delta x$$ and $$\Delta y = 0$$.

Thus $$\frac{(z + \Delta z) - f(z)}{\Delta z}$$

= $$\frac{u(x +\Delta x, y) + iv(x + \Delta x, y) - u(x,y) - iv(x,y)}{\Delta x}$$

= $$\frac{u(x + \Delta x,y) - u(x,y)}{\Delta x}$$ + i $$\frac{v(x +\Delta x,y)-v(x,y)}{\Delta x}$$

Now

$$\lim_{\Delta z \rightarrow 0} \frac{f(z+\Delta z)-f(z)}{\Delta z}$$

= $$\lim_{\Delta x \rightarrow 0} \frac{u(x,y + \Delta y) - u(x,y)}{\Delta x}$$ + i  $$\lim_{\Delta x \rightarrow 0} \frac{v(x + \Delta x,y) - v(x,y)}{\Delta x}$$

Thus  f'(z) = $$\frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}$$  .........(1)

Case II. Next let $$\Delta z$$ approaches zero along the imaginary axis (or y-axis), then $$\Delta z$$ = i$$\Delta y, \Delta x = 0$$ and we have

$$\frac{f(z + \Delta z) - f(z)}{\Delta z}$$

= $$\frac{u(x, y+\Delta y) + iv(x, y+\Delta y) - u(x,y) - iv(x,y)}{i\Delta y}$$

= $$\frac{u(x, y+ \Delta y) - u(x,y)}{i\Delta y}$$ + $$\frac{v(x, y+\Delta y)-v(x,y)}{\Delta y}$$

Thus

f'(z) = $$\lim_{\Delta z \rightarrow 0} \frac{f(z+\Delta z)-f(z)}{\Delta z}$$

= $$\lim_{\Delta y \rightarrow 0} \frac{u(x,y + \Delta y) - u(x,y)}{i\Delta y}$$ +   $$\lim_{\Delta y \rightarrow 0} \frac{v(x, y+ \Delta y) - v(x,y)}{\Delta y}$$

= $$\frac{1}{i} \frac{\partial u}{\partial y} + \frac(\partial v}{\partial y}$$

$$\therefore f'(z)$$= $$\frac{\partial v}{\partial y} - i \frac{\partial u}{\partial y)$$   .........(2)

From (1) and (2), we have

f'(z) = \(\frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}

= \(\frac{\partial v}{\partial y} + i \frac{\partial u}{\partial y}

Equating real and imaginary parts in above, we obtain

\(\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \frac{\partial v}{\partial x} = \frac{-\partial u}{\partial y}  ............(3)

Equations (3), known as the  Cauchy-Riemann equations give the necessary condition for a function f(z) to be analytic.