Results 1 to 3 of 3

Math Help - complex integral

  1. #1
    Junior Member
    Joined
    May 2009
    Posts
    41

    complex integral

    How we solve those?
    Attached Thumbnails Attached Thumbnails complex integral-complex-integral.jpg  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    May 2009
    Posts
    471

    example of Cauchy's Integral Formula

    For the first one, if n\geq0, then we can use Cauchy's Theorem to change the curve to a point through a homotopy and the integral over a point is zero.

    For n=-1 use Cauchy's Integral formula to conclude the answer is 2\pi\imath

    For n\leq-2 then we have to use Cauchy's General Integral Formula which is f^n(a)=\frac{n!}{2\pi\imath}\int_C\frac{f(z)}{(z-a)^{n+1}} where f^n(a) is the n-th derivative of f(z) evaluated at the point a

    So...

    \int_C(z-a)^ndz=\int_C\frac{1}{(z-a)^k}dz for k=-n and by Cauchy's Integral formula this integral is

    \frac{2\pi\imath}{(k-1)!}f^{(k-1)}(z) but f(z)=1 in this case, so the k-1 derivative of 1 is 0 (as k-1 is always at least 1) and anything times 0 is 0.
    Last edited by artvandalay11; May 25th 2009 at 07:35 PM. Reason: make it more clear
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    May 2009
    Posts
    471
    \oint_C\frac{z^3+z}{(2z+1)^3}dz=\oint_C\frac{z^3+z  }{(2(z+\frac{1}{2}))^3}dz=\oint_C\frac{z^3+z}{2^3(  z+\frac{1}{2})^3}dz

    =\oint_C\frac{\frac{z^3+z}{8}}{(z+\frac{1}{2})^3}d  z

    Now Cauchy's integral formula states:

    f^n(a)=\frac{n!}{2\pi\imath}\oint_C\frac{f(z)}{(z-a)^{n+1}} where f^n(a) is the n-th derivative of f(z) evaluated at the point a

    So =\oint_C\frac{\frac{z^3+z}{8}}{(z+\frac{1}{2})^3}d  z=\frac{2\pi\imath}{2!}*f ''(-\frac{1}{2}) where f(z)= \frac{z^3+z}{8}

    I'll leave it to you to find the second derivative, plug in z= \ -\frac{1}{2} and multiply it by \frac{2\pi\imath}{2!}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. complex integral
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: September 6th 2011, 12:26 PM
  2. [SOLVED] Complex Integral in Complex Analysis
    Posted in the Advanced Math Topics Forum
    Replies: 1
    Last Post: August 21st 2011, 10:46 PM
  3. Complex Integral
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 3rd 2010, 05:30 PM
  4. Complex integral help
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: March 20th 2009, 10:48 AM
  5. complex integral
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 21st 2008, 06:31 AM

Search Tags


/mathhelpforum @mathhelpforum