Question

Describe Cauchy Riemann Equations.

Answer 1

Hint : Here in this question we have to describe the Cauchy Riemann equations. Usually the Cauchy Riemann Equations are based on Analytic functions that are Complex number, Continuity and differentiability. Hence we get the required solution for the given question.

Complete step-by-step answer :
The Cauchy Riemann equations is a pair of real-valued functions of two real variables $u(x,y)$and $v(x,y)$
That is $\dfrac{{\partial u}}{{\partial x}} = \dfrac{{\partial v}}{{\partial y}}$ $\to$(1)
$\dfrac{{\partial u}}{{\partial y}} = - \dfrac{{\partial v}}{{\partial x}}$ $\to$(2)
Analytic function states that
Function f(z) is said to be analytic at a point $z = {z_0}$ if it is differentiable not only at ${z_0}$ but neighbourhood of ${z_0}$. Necessary condition for a function f(z) to be analytic.
Statement: If f(z) is a analytic in a domain D, the partial ${U_x}$, ${U_y}$, ${V_x}$, ${V_y}$ exits and satisfy
${U_x}$=${V_y}$ and ${U_y}$= -${V_x}$ (from 1 and 2) where $f(z) = u(x,y) + iv(x,y)$
Suppose if we consider Cauchy-Riemann equations in polar form:
\dfrac{{\partial u}}{{\partial r}} = \dfrac{1}{r}$$$$\dfrac{{\partial v}}{{\partial \theta }} and $\dfrac{{\partial u}}{{\partial \theta }}$= - r$\dfrac{{\partial v}}{{\partial r}}$
Now to know the detail explanation of Cauchy Riemann equations
Consider, when a complex function $f(z) = u + i\,v$ is a complex differentiable. If the complex derivative $f'(z)$ is to exist, then we should be able to compute it by approaching z along either horizontal or vertical lines.
Thus we must have

$$\Rightarrow f'(x) = \mathop {\lim }\limits_{t \to 0} \dfrac{{f(z + t) - f(z)}}{t} \\\ \Rightarrow f'(x) = \mathop {\lim }\limits_{t \to 0} \dfrac{{f(z + it) - f(z)}}{{it}} \;$$

Where $t$ is a real number and $i$ is an imaginary number.
In terms of u and v,

$$\Rightarrow \mathop {\lim }\limits_{t \to 0} \dfrac{{f(z + t) - f(z)}}{t} \\\ \Rightarrow \mathop {\lim }\limits_{t \to 0} \dfrac{{u(x + t,y) + i\,v(x + t,y) - u(x,y) - v(x,y)}}{t} \;$$ $$\Rightarrow \mathop {\lim }\limits_{t \to 0} \dfrac{{u(x + t,y) - u(x,y)}}{t} + \mathop {\lim }\limits_{t \to 0} \dfrac{{v(x + t,y) - v(x,y)}}{t} \\\ \Rightarrow \dfrac{{\partial u}}{{\partial x}} + i\dfrac{{\partial v}}{{\partial x}} \;$$

Taking the derivative along a vertical line

$$\Rightarrow \mathop {\lim }\limits_{t \to 0} \dfrac{{f(z + it) - f(z)}}{{it}} \\\ \Rightarrow \mathop {\lim }\limits_{t \to 0} \dfrac{{u(x,y + t) + iv(x,y + t) - u(x,y) - v(x,y)}}{t} \\\ \Rightarrow - i\,\mathop {\lim }\limits_{t \to 0} \dfrac{{u(x,y + t) - u(x,y)}}{t} + \mathop {\lim }\limits_{t \to 0} \dfrac{{v(x,y + t) - v(x,y)}}{t} \\\ \Rightarrow - i\dfrac{{\partial u}}{{\partial y}} + \dfrac{{\partial v}}{{\partial y}} \;$$

Equating real and imaginary parts,
If a function $f(z) = u + i\,v$ is complex differentiable, then its real and imaginary parts satisfy the Cauchy Riemann equations:
$\Rightarrow \dfrac{{\partial u}}{{\partial x}} = \dfrac{{\partial v}}{{\partial y}}$
$\Rightarrow \dfrac{{\partial u}}{{\partial y}} = - \dfrac{{\partial v}}{{\partial x}}$
The complex derivative $f'(z)$ is given by
$f'(z) = \dfrac{{\partial u}}{{\partial x}} + i\dfrac{{\partial v}}{{\partial x}} = \dfrac{{\partial v}}{{\partial y}} - i\dfrac{{\partial u}}{{\partial y}}$

Note : Continuity function: A function f(z) is said to be continuous at $z = {z_0}$ if $\mathop {\lim }\limits_{z \to {z_0}} f(z) = f({z_0})$
Differentiability: A function f(z) is said to be differentiable at $z = {z_0}$ if $\mathop {\lim }\limits_{z \to {z_0}} \dfrac{{f(z) - f({z_0})}}{{z - {z_0}}}$ exists and $f'({z_0})$.
Complex number: An ordered pair of reals is called as complex number written as (a, b) where a, b$\in$R represented by $z = a + ib$, $i = \sqrt { - 1}$ $\forall$ a , b $\in$R $C = \\{ a + ib\,:\,\,a,b \in R,\,\,i = \sqrt { - 1} \\}$