Results 1 to 3 of 3

Thread: Fundamental Theorem of Complex Calculus

  1. #1
    Senior Member slevvio's Avatar
    Joined
    Oct 2007
    Posts
    347

    Fundamental Theorem of Complex Calculus

    Hello everyone, I was having trouble with the proof of this and I was wondering if anyone could help me!

    Let $\displaystyle f:S \rightarrow \mathbb{C}$ be analytic (where S is an open set), and such that if $\displaystyle \gamma : [a,b] \rightarrow S$ is a piecewise continuously differentiable path in S, then if f' is continuous on $\displaystyle \gamma^* $(which is always true anyway), and$\displaystyle z_1:= \gamma (a), z_2:= \gamma (b),$

    then $\displaystyle \int_{\gamma} f'(z)dz = f(z_2)-f(z_1)$

    PROOF: Write $\displaystyle \gamma $as $\displaystyle \gamma_1 + \gamma_2 + \ldots + \gamma_n$, where each $\displaystyle \gamma_i : [t_{i-1}, t_i] \rightarrow S$ is a simple smooth path, with $\displaystyle \gamma_i (t_i) = \gamma_{i+1} (t_i), \gamma_1 (t_1) := \gamma(a), \gamma_n (t_n) := \gamma (b).$

    Then $\displaystyle \int_{\gamma_i} f'(z) dz = \int_{t_{i-1}}^{t_i} f '(\gamma_i (t) ) \gamma_i ' (t) dt = \int_{t_{i-1}}^{t_i} \dfrac{d}{dt} f ( \gamma_i (t) ) dt = $ now here I have a problem because it says this line is equal to

    $\displaystyle
    \left[ f(\gamma_i (t)) \right]_{t_{i-1}}^{t^i}
    $ by the fundamental theorem of calculus for REAL functions. however $\displaystyle f \circ \gamma_i $ is not real valued is it? so why can i use it here?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2008
    From
    Paris, France
    Posts
    1,174
    Just apply the fundamental theorem of calculus to the real and imaginary parts of the function. Indeed, $\displaystyle \int \phi(t)dt=\int {\rm Re}(\phi(t))dt+i\int {\rm Im}(\phi(t))dt$ for $\displaystyle \phi:\mathbb{R}\to\mathbb{C}$ (and a function is differentiable iff both its real and imaginary parts are). Similarly, most theorems like integration by part or change of variables adapt seamlessly to complex-valued functions.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member slevvio's Avatar
    Joined
    Oct 2007
    Posts
    347
    ah I see this now! thanks very much
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Using the Fundamental Theorem of Calculus...
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Aug 5th 2009, 08:09 AM
  2. Fundamental Theorem of Calculus
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Apr 15th 2009, 07:24 PM
  3. FUNdamental Theorem of calculus
    Posted in the Calculus Forum
    Replies: 6
    Last Post: Jan 28th 2009, 03:59 PM
  4. The Fundamental Theorem of Calculus
    Posted in the Calculus Forum
    Replies: 10
    Last Post: Nov 17th 2008, 03:48 PM
  5. Replies: 2
    Last Post: Jun 14th 2007, 06:35 AM

Search Tags


/mathhelpforum @mathhelpforum