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

[BUConsul]

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, f_x= 0, f_y=1 By Cuachy-Reimann we have that f'(z) exists for all points that satisfy f_x=-if_y 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: "didifferentiable 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.

 

[MathematicalPhysicist]

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.

 

[micromass]

Yeah, mathematicalphysicist is correct. The book is wrong I think...
Can you tell us what book you are using?

 

[BUConsul]

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?

 

[snipez90]

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.