Cauchy Riemann Equations For Analytic Function

Cauchy - Riemann Equations :-

Cauchy-Riemann Equation

Necessary Conditions For A Function To Be Analytic :

Cauchy-Riemann Equation

Statement:-

Cauchy-Riemann Equation

The necessary conditions for

w = f(z) = u(x,y) + i v(x,y) to be analytic (differentiable) at any point z = x + i y of its domain D is that the four patial derivatives ∂u/∂x , ∂u/∂y ,∂v/∂x ,∂v/∂y should exists and satisfy the C-R partial differential equations .

∂u/∂x = ∂v/∂y and ∂u/∂y = - ∂v/∂x

Proof :-

Cauchy-Riemann Equation

Let f(z) = u(x,y) + i v(x,y) be analytic at any point z of its domain , then

f'(z) = lim f(z+δz) - f(z) / δz

δx-->0

exists and is unique .

i.e it is independent of the path along which δz -->0

Let z = x+ iy ∴ δz = δx + i δy and as δz-->0

then δx , δy --> 0

∴ f'(z) =

lim [u(x+δx ,y+δy) + iv(x+δx,y+δy)]/δx+iδy

δx-->0

δy-->0

lim [u(x,y) + iv(x,y)] / δx+iδy

δx-->0

δy-->0

lim [u(x+δx , y+δy) - u(x,y)] / δx+iδy

δx-->0

δy-->0

lim i [v(x+δx , y+δy) - v(x,y)] / δx + i δy

δx-->0

δy-->0 .......(1)

Now let us consider two possible approaches in which δz -->0

Case-I:-

Cauchy-Riemann Equation

Let δz be purely real , then δz = δx , δy=0 and δx -->0

Hence from (1) , we get

f'(z) = lim [u(x+δx ,y) - u(x,y)]/δx

δx-->0

lim i[v(x+δx , y) - V(x,y)]/ δx

δx-->0

or f'(z) = ∂u/∂x + i ∂v/∂x ..........(2)

Since f'(z) exists , then above limit exists i.e. ∂u/∂x and ∂v/∂x exists .

Case-II:-

Cauchy-Riemann Equation

Let δz-->0 along the imaginary axis , then δz is purely imaginary and then δx = iδy , δx= 0 and δy-->0 . Hence from(1) we get

f'(z) = lim [u(x, y+iy) - u(x,y)]/iy

δy-->0

lim i [v(x,y+δy) - u(x,y)]/iδy
δy-->0

or, f'(z) = 1/i ∂u/∂y + ∂v/∂y

or, f'(z) = -i ∂u/∂y + ∂v/∂y .........(3)

Since f'(z) exists ,

then above limit exists i.e. ∂u/∂y abd ∂v/∂y exists .

Now , the two limits in (2) and (3) are identical and unique .

∂u/∂x + i ∂v/∂x = -i ∂u/∂y + ∂v/∂y

Equating real and imaginary parts we get

∂u/∂x = ∂v/∂y ........(4)

and ∂u/∂y = -∂v/∂x ........(5)

Equations (4) and (5) are called Cauchy - Riemann Partial Differential Equations .

Polar Form Of Cauchy - Riemann Equations :

Cauchy-Riemann Equation

If w = f(z) = u+iv is an analytic function, show that , in polar form , the C - R equations are :

∂u/∂r = 1/r ∂v/∂θ , ∂v/∂r = -1/r ∂u/∂θ

Proof :-

Cauchy-Riemann Equation

We know that C-R equations are

∂u/∂x = ∂v/∂y and ∂u/∂y = - ∂v/∂x ...(1)

Let x = r cos θ , y= r sin θ

So (x²+y²) = r² ⇒√(x²+y²) = r

Now y/x = tan θ ⇒θ = tan⁻¹(y/x)

Then ∂r/∂x = 1/2 .1/√(x²+y² ) .2x = x/√(x²+y²)

= x/r = r cos θ /r = cos θ

Similarly , ∂r/∂y = y/r = r sin θ /r = sin θ

Again θ = tan⁻¹(y/x)

∂θ/∂x = 1/(1+y²/x²) (-y/x²) = -y/x²+y²

= -r sin θ /r² = -sin θ /r

∂θ/∂y = 1/(1+y²/x²) (1/x) = x/x²+y²

= r cos θ / r² = cos θ /r

∴ ∂u/∂x = ∂u/∂r . ∂r/∂x + ∂u/∂θ . ∂θ/∂x

= cos θ . ∂u/∂r - sin θ / r .∂u/∂θ

∂v/∂y = ∂v/∂r . ∂r/∂y + ∂v/∂θ .∂θ/∂y

= sin θ . ∂v/∂r + cos θ /r . ∂v/∂θ

From (1) , we get

∂u/∂x = ∂v/∂y

⇒cos θ ∂u/∂r - sin θ /r ∂u/∂θ

= sin θ ∂v/∂r + cos θ /r ∂v/∂θ
..........(2)

∂v/∂x = ∂v/∂r . ∂r/∂x + ∂v/∂θ . ∂θ/∂x

= cos θ. ∂v/∂r - sin θ /r .∂v/∂θ

∂u/∂y = ∂u/∂r . ∂r/∂y + ∂u/∂θ . ∂θ/∂y

= sin θ. ∂u/∂r + cos θ / r .∂u/∂θ

Again from (1) , we get

-∂v/∂x = ∂u/∂y

⇒sin θ /r . ∂v/∂θ - cos θ .∂v/∂r

= sin θ .∂u/∂r + cos θ /r .∂u/∂θ
............(3)

Multiplying equation (2) by cos θ and equation (3) by sin θ , we get

cos² θ . ∂u/∂r - sin θ . cos θ /r . ∂u/∂θ

= sin θ .cos θ ∂v/∂r + cos² θ /r . ∂v/∂θ

..............(4)

sin² θ . ∂u/∂r + cos θ sin θ /r . ∂u/∂θ

= -cos θ .sin θ ∂v/∂r + sin² θ /r . ∂v/∂θ

...............(5)

Adding equation(4) with equation (5), we get

(cos² θ + sin² θ).∂u/∂r

= (cos² θ + sin² θ)/r . ∂v/∂θ

⇒∂u/∂r = 1/r . ∂v/∂θ .............(6)

Multiplying equation (2) by sin θ and equation (3) by cos θ , we get

cos θ . sin θ . ∂u/∂r - sin² θ /r . ∂u/∂θ

= sin² θ . ∂v/∂r + sin θ . cos θ /r . ∂v/∂θ

...........(7)

sin θ . cos θ . ∂u/∂r + cos² θ /r . ∂u/∂θ

= -cos² θ . ∂v/∂r + cos θ.sin θ /r . ∂v/∂θ

............(8)

Subtracting we get ,

-(sin² θ + cos² θ)/r . ∂u/∂θ

= (sin² θ + cos² θ).∂v/∂r

⇒-1/r . ∂u/∂θ = ∂v/∂r

⇒∂v/∂r = -1/r . ∂u/∂θ ................(9)

So , equation(6) and equation (8) are called C-R equations in polar form .

Cauchy-Riemann Equation

About Scientists :

Cauchy-Riemann Equation

Baron Augustin-Louis Cauchy (21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics.For more .

Cauchy-Riemann Equation

Georg Friedrich Bernhard Riemann was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. For more.

Cauchy-Riemann Equation

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