Derivative of a complex conjugate?

In summary, the conversation discusses a problem involving a complex conjugate and its derivative. The student initially assumes that the complex conjugate is a constant, but after substitution, realizes that they are missing something. The instructor explains that the complex conjugate of exp(ikx) is exp(-ikx) and reminds the student that the complex conjugate of C is C^*. The correct derivative is ik*exp(ikx).
  • #1
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Homework Statement



[PLAIN]http://img823.imageshack.us/img823/4500/85131172.png [Broken]

Homework Equations



Derivations and substitutions.

The Attempt at a Solution



Basically it seems like a very simple problem to me however I can't seem to get the right answer. First I just assumed that the c.c. (complex conjugate) was just a constant thus:

[tex]\Psi[/tex]'(x) = Ckeikx
[tex]\Psi[/tex]''(x) = Ck2eikx

But substituting that equation into the original DE gives:

Ck2eikx = Ck2eikx + k2(c.c)

obviously I'm missing something.

edit: maybe I read the question wrong could c.c. mean Ce-ikx ?
 
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  • #2
Yes, the complex conjugate of exp(ikx) is exp(-ikx). And don't forget the c.c. of C is C^*. And d/dx(exp(ikx)) isn't k*exp(ikx) either, it's ik*exp(ikx).
 
  • #3
Ok, I understand now. Also thanks for pointing out my derivative mistake!
 

1. What is the definition of a complex conjugate?

The complex conjugate of a complex number is another complex number with the same real part but an opposite imaginary part. For example, the complex conjugate of 3+2i is 3-2i. It is denoted by adding a bar over the number, such as z̅.

2. What is the derivative of a complex conjugate?

The derivative of a complex conjugate is the complex conjugate of the derivative of the original complex number. In other words, if f(z) is a complex function, then the derivative of its complex conjugate, f(z̅), is equal to the complex conjugate of the derivative of f(z), denoted by f'(z̅).

3. How is the derivative of a complex conjugate calculated?

The derivative of a complex conjugate is calculated using the standard rules of differentiation, but with the added step of taking the complex conjugate at the end. This means taking the derivative of the real and imaginary parts separately, and then combining them with the opposite sign for the imaginary part.

4. What is the relationship between the derivative of a complex number and its complex conjugate?

The derivative of a complex number and its complex conjugate are related by the fact that they both have the same real part, but opposite imaginary parts. This means that their derivatives will also have the same real part, but opposite imaginary parts. In other words, the derivative of a complex conjugate is the complex conjugate of the derivative.

5. Why is the derivative of a complex conjugate important in complex analysis?

The derivative of a complex conjugate is important in complex analysis because it allows us to find the derivatives of complex functions that are not differentiable in the traditional sense. By taking the complex conjugate, we can find the derivative of the original function at points where it may not be defined or may have a discontinuity. This is useful in solving problems and understanding the behavior of complex functions.

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