Cauchy-riemann using polar co-ordinates

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In summary, the function f(z) = U(x,y) + iV(x,y) is given, where U(x,y) = y / ((1-x)^2 + y^2). To check if it is harmonic, the author suggests using polar coordinates for easier differentiation. The logarithm of the function is considered and the second derivatives are calculated. However, u_{xx} + u_{yy} is not equal to 0, so the function is not harmonic. The Cauchy Rienman conditions are then used to determine V(x,y) and the final result is f(z) = -i / (1-z).
  • #1
randybryan
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I've been given a function U(x,y) where f(z) = U(x,y) + iV(x,y)

and asked to check if it is harmonic and then work out what V(x,y) is

U(x,y) = [tex]\frac{y}{(1 - x)^{2} + y^{2}}[/tex]

To check if it is harmonic I can see if d2U/dx2 + d2U/dy2 = 0

I've tried differentiating and it's fairly arduous, so I'm thinking it might be easier to use polar co-ordinates (As a lot of the other questions simplify when doing this). Can anyone think of a suitable substitution to make the differentiation easier?

the answer in the back of the book says that f(z) = -i / (1 - z)
 
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  • #2
I suggest you consider the logarithm of the function.
log u = log y + log((1-x)2+y2)

[tex]\frac{1}{u}\frac{\partial u}{\partial y}=\frac{1}{y}+\frac{2y}{(1-x)^2+y^2}[/tex]

[tex]\frac{\partial u}{\partial y}=\frac{1}{(1-x)^2+y^2}+2u^2[/tex]
 
  • #3
Thanks, but I still don't know where to go from here
 
  • #4
So its not helping after all.
I give a try for the second derivatives and obtain

[tex]u_{yy}=\frac{2u^2}{y}+8u^3[/tex] and [tex]u_{xx}=\frac{-2u^2}{y}+\frac{8(1-x)^2u^3}{y^2}[/tex]

so that
[tex]u_{xx}+u_{yy}=\frac{8u^2}{y}\neq 0 [/tex] ?
 
  • #5
TRY this
for a function to be harmonic its laplacian should be zero
so [tex]\nabla[/tex]2U = 0

( i worked it out and it is true)

THEN use the Cauchy Rienman conditions to determine v

Finally the f(z)=U+iV
 

1. What is the Cauchy-Riemann equation in polar coordinates?

The Cauchy-Riemann equation in polar coordinates is a set of two equations that express the relationship between the real and imaginary parts of a complex-valued function. It is written as: r(r, θ) = f(r, θ) + ig(r, θ), where r is the radial distance from the origin and θ is the angular position.

2. How is the Cauchy-Riemann equation used in polar coordinates?

The Cauchy-Riemann equation is used to determine if a complex-valued function is analytic. If a function satisfies the equations, then it is said to be analytic in the region where the equations hold. This means that the function is differentiable at every point in that region and has a unique value for its derivative.

3. Can the Cauchy-Riemann equation be expressed in terms of derivatives?

Yes, the Cauchy-Riemann equation can be expressed in terms of derivatives. In polar coordinates, the partial derivatives of the real and imaginary parts of a function are used to represent the Cauchy-Riemann equations. This allows for the determination of analyticity by checking if the derivatives of the function satisfy the equations.

4. What is the significance of the Cauchy-Riemann equation in complex analysis?

The Cauchy-Riemann equation is a fundamental tool in complex analysis. It is used to determine if a function is analytic, which is necessary for many of the important theorems and techniques in complex analysis. It is also used to find the derivative and other properties of complex functions, making it an essential tool in understanding the behavior of complex-valued functions.

5. Are there any limitations or restrictions when using the Cauchy-Riemann equation in polar coordinates?

One limitation of using the Cauchy-Riemann equation in polar coordinates is that it can only be applied to functions that are differentiable at every point in a given region. This means that functions with singularities or discontinuities may not satisfy the equations. Additionally, the equations may not hold in certain regions where the function is not defined, such as at the origin.

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