Expert: Paul Klarreich - 9/30/2011

Question
please help me out with this question

Q) Using the definition of the derivative as a limit, give a direct proof that dw/dz = −2/z3
when w = 1/z2

Answer
Questioner:Attir
Country:Ontario, Canada
Category:Advanced Math
Private:No
Subject:COMPLEX VARIABLE

Question:

Q) Using the definition of the derivative as a limit, give a direct proof that dw/dz = −2/z3
when w = 1/z2

..............................
I may be missing something, but I think the proof goes something like this:

(same as the real-variables proof?)
        1
w(z) = ----,  assuming  z /= 0
       z^2

        1
w(r) = ----
       r^2

(r is also a complex variable.)

w(r) - w(z)
----------- =
 r - z

1/r^2 - 1/z^2
-------------- =
 r - z
[after multiplying through by r^2 z^2 ]

z^2 - r^2
-------------- =
r^2 z^2(r - z)

(z - r)(z + r)
-------------- =
r^2 z^2(r - z)

-(r - z)(z + r)
---------------- =
r^2 z^2(r - z)

- (z + r)
----------
r^2 z^2

Now at  lim  r --> z, this becomes:

- (z + z)
---------- =
z^2 z^2

- 2z
----- =
z^4

-2
----
z^3