Cauchy-Riemann Equations

Cauchy-Riemann equation is a powerful tool for checking analyticity of a complex-valued function and is considered to be the one of pillars on which complex analysis rests. 

Roughly speaking,f is analytic in a domain D or region R if and only if the first-order partial derivatives of u and v satisfy the two so called Cauchy-Riemann equations

i.e                     `U_x = V_y`          ,             `U_y = - V_x`

or equivalently,

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

everywhere in region R or domain D.

Examples: 

check the analyticity of the functions.

1.   `f(z) = z^2 = x^2 - y^2 + 2ixy` 

here,    `u(x, y) = x^2 - y^2`    ,   `v(x, y) = 2xy`

`u_x = 2x`    ,    `v_x = 2y`   ,    `u_y = - 2y`  ,   `v_y = 2x`

clearly   `u_x = 2x = v_y`    ,     `v_x = 2y = - u_y`

hence f(z) is analytic function.

2.   f(z) = 𝑧̅ = x - iy

here,    u(x, y) = x   ,   v(x, y) = -y

`u_x = 1     ,     u_y = 0    ,     v_x = 0     ,     v_y = -1`

`u_x = 1 \neq -1 = v_y`    ,     `v_x = 0 = u_y = 0`

Although second Cauchy-Riemann equation is satisfied but the first equation is not true. Hence the given function is non-analytic function everywhere in complex plane.

3.    `f(z) = z^3 = x^3 - iy^3 + 3ixy(x + iy)`

`u(x, y) = x^3 - 3xy^2    ,      v(x, y) = -y^3 + 3x^2 y`

`u_x = 3x^2 - 3y^2    ,     v_x = 6xy`

`u_y = -6xy     ,      v_y = -3y^2 + 3x^2`

hence,    `u_x = 3x^2 - 3y^2 = v_y      ,     u_y = -6xy = - v_x`

so, the CR- equations are satisfied. therefore the given function is analytic everywhere.

Polar form of Cauchy Riemann equations

Now we discussed the polar form of Cauchy Riemann equations if

`z = r(\cos(\theta) + \sin(\theta))`  then, obviously

`f(z) = u(r, \theta) + iv(r, \theta)`

then Cauchy Riemann equations become

`u_r = \frac{1}{r} v_\theta     ,     v_r = - \frac{1}{r} u_\theta`

Examples: