Integrating the Complex conjugate of z with respect to z

In summary, the conversation is about a question related to contour integration and line integration involving a complex number and its conjugate. The question asks to evaluate the integral over a line segment from -i to 1+i. The expert advises to treat it as a complex line integral and parametrize the line as a function of t. They also mention the lack of an antiderivative for the function and suggest taking a break before attempting to solve the problem. The conversation ends with the expert thanking the person for their prompt responses.
  • #1
Deevise
5
0
Im doing a bit of contour integration, and a question came up with a term in it am unsure of how to do: in its simplest form it would be

[tex]\int[/tex][tex]\bar{z}[/tex]dz

where z is a complex number and [tex]\bar{z}[/tex] is it's conjugate. Hmm i can't get the formatting to work out properly.. :S
 
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  • #2
If you are integrating over a circular contour of radius R then zz*=R^2, so z*=R^2/z. Otherwise you just have to take the contour and write it as z=(x(t)+iy(t)), so z*=(x(t)-iy(t)).
 
  • #3
Well now i feel kind of stupid... its line intergration, not contour integration :P the question reads:

Evaluate the integral:

[tex]\int[/tex]( [tex]\bar{z}[/tex] +1 ) dz
L

Where L is the line segment from -i to 1+i.

normally i would just integrate and sub in start and end point, but i have totaly drawn a blank on what to do with the conjugate in this case...
 
  • #4
Just treat it as a complex line integral. You can only 'sub in' endpoints if the function you are integrating is analytic and has an antiderivative. (z*+1) doesn't. Parametrize L as a function of t and integrate dt. Like I said, if you have z=(x(t)+iy(t)) then z*=(x(t)-iy(t)).
 
  • #6
I think it's time i went to sleep... Yeh now that you mention the lack of anti-derivative i knew that. I think a good nights sleep will prepare me better for this exam than grinding my head into non-exsistant problems...

sorry to waste your time with inane questions lol... Thanks for the prompt responces.
 

FAQ: Integrating the Complex conjugate of z with respect to z

What does it mean to integrate the complex conjugate of z with respect to z?

Integrating the complex conjugate of z with respect to z means finding the antiderivative of the complex conjugate function with respect to the complex variable z. This involves finding a function whose derivative is equal to the complex conjugate function.

What is the purpose of integrating the complex conjugate of z?

The purpose of integrating the complex conjugate of z is to solve complex integrals and evaluate complex functions in terms of simpler functions. It can also be used to calculate areas and volumes in the complex plane.

What are the steps involved in integrating the complex conjugate of z?

The first step is to identify the complex function and its corresponding complex conjugate. Then, use the properties of integration to simplify the complex function. Next, find the antiderivative of the complex conjugate function. Finally, add a constant of integration to the result.

Can the complex conjugate of z be integrated using the same techniques as real numbers?

Yes, the integration techniques used for real numbers can also be applied to the complex conjugate of z. However, complex integrals may require additional techniques, such as using the Cauchy-Riemann equations or contour integration.

Are there any specific applications of integrating the complex conjugate of z in science?

Yes, integrating the complex conjugate of z has various applications in physics, engineering, and mathematics. It is commonly used in electromagnetic theory, signal processing, and quantum mechanics to solve complex problems and model physical systems.

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