Complex Derivative of Im(z): Find f'(z)

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In summary: Overall, it seems like you are doing well with your pre-studying for your upcoming Complex Variables/Analysis class. You have learned about complex derivatives, the Caughy-Riemann equations, harmonivs functions, diffrentiability, and holomorphicity. However, you struggled with one of the problems in the back of the chapter and sought help from others. You discovered that the solution in the book was incorrect and that there is no limit for the derivative at z=0 for f(z)=Im(z). You also received recommendations for a good textbook for complex analysis. In summary, you are doing well with your pre-studying and have a good understanding of the concepts covered so far.
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BUConsul
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So, Iv'e been doing a little bit of prestudying for the Complex Variables/Analysis calss that I'm taking in the Spring and Iv'e just finished learning about complex derivatives. So, the Caughy-Riemann equations harmonivs functions and the like. I also learned about diffrentiability (in the Complex Plane) and holomorphicity. So I was doing the HW in the back of the chapter and I wasn't able to do one of the problems.

Find if f'(z) exists, and if it does state at which points is it differntiable and where is it holomorphic and find f'(z)
I. f(z)=Im(z )

I. f(z)=Im(z)=f(x,y)=Im(x+iy)=y, fx= 0, fy=1 By Cuachy-Reimann we have that f'(z) exists for all points that satisfy fx=-ify Subbing in: 0 = -i(1) -> Since there exists no x,y in R that satisfies the given equations f(z) is nowhere differentiable. But the solution in the back of the book says: "di differentiable at 0 with derivative 0, nowhere holomorphic". I'm a bit confused here, and I would greatly appreciate any help, I noticed that f:C -> R and was wondering if that had anything to do with the solution.
 
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  • #2
Well if you calculate the derivative at z=0 you get:

Im z / z = y/(x+iy)=y(x-iy)/(x^2+y^2)

if you first calculate x->0 and afterwards y->0 you get -i.
If first y->0 and then x->0 you get 0.

So no there isn't a limit in z=0.
 
  • #3
Yeah, mathematicalphysicist is correct. The book is wrong I think...
Can you tell us what book you are using?
 
  • #4
I'm not using any book, perse. Instead I'm lecture notes as found here: http://www.math.binghamton.edu/sabalka/teaching/09Spring375/complex.pdf . Thanks for the help, MathematicalPhysicist and MicroMass, glad to know I'm not completely mis understanding the concept. By the way, does anyone know of a better text for complex variables?
 
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  • #5
Judging by a quick perusal, that set of lecture notes is not bad, and it's pretty generous for the authors to give problems and solutions, even if there was apparently an oversight. I don't see any problem with continuing to work through it. Also wikipedia is good for complex analysis topics, and I completely forgot about looking there when I took complex analysis this past quarter (my teacher had good lecture notes, but wikipedia's articles have some neat applications pulled from various sources).

If you want a textbook, I recommend Ahlfors Complex Analysis, especially if you prefer a geometric point of view.
 

1. What is the definition of a complex derivative?

The complex derivative of a function f(z) is defined as the limit of the difference quotient as the change in z approaches 0, where z is a complex number.

2. How do you find the complex derivative of a function?

The complex derivative of a function f(z) can be found by applying the Cauchy-Riemann equations, which relate the partial derivatives of the real and imaginary parts of f(z) to its complex derivative.

3. What is the relationship between the complex derivative and the Cauchy-Riemann equations?

The Cauchy-Riemann equations provide a necessary condition for a complex function to be differentiable. If the partial derivatives of the real and imaginary parts of a function satisfy these equations, then the function is complex differentiable and its complex derivative can be found using these equations.

4. Can the complex derivative of a function exist at a point but not be continuous?

Yes, it is possible for a function to be complex differentiable at a point but not continuous at that point. This can occur if the function satisfies the Cauchy-Riemann equations at that point, but the limit of the difference quotient does not exist.

5. How is the concept of a complex derivative used in real-world applications?

The concept of a complex derivative is used in various fields of science and engineering, such as fluid mechanics, electromagnetics, and signal processing. It allows for the analysis and modeling of complex systems, and is also used in the development of advanced mathematical techniques and algorithms.

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