Results 1 to 3 of 3

Math Help - Complex Power Series

  1. #1
    Junior Member
    Joined
    Apr 2010
    Posts
    56

    Complex Power Series

    Question:

    Let F(z) be the antiderivative of f(z)=e^(z^2)with F(0). Find F(z) as a power series around z=0.

    Attempt at solution:

    We know that, e^z = \sum_{n=0}^{\infty} \frac{1}{n!}z^n

    Hence, e^{z^2} = \sum_{n=0}^{\infty} \frac{1}{n!}z^{2n}

    Thus,

    F(z)=\int e^{z^2} = C+\sum_{n=0}^{\infty} \frac{1}{(2n+1)n!}z^{2n+1}

    Now this is where i have problems, to find C,

    let e^{z^2}= (cos\theta +isin\theta)^2

    Integrating we find,

    \int(cos\theta +isin\theta)^2 = \frac{-1}{2}icos^2\theta+\frac{1}{2} sin2\theta=F(z)

    Therefore,

    F(z)= \frac{-1}{2}icos^2\theta+\frac{1}{2} sin2\theta=C+\sum_{n=0}^{\infty} \frac{1}{(2n+1)n!}z^{2n+1}

    => F(0)=\frac{-i}{2}= C

    Therefore,

    F(z)= \frac{-i}{2} +\sum_{n=0}^{\infty} \frac{1}{(2n+1)n!}z^{2n+1}

    Is my methodology correct? there is no answer in the book for some reason, but upon research i think my power series is correct, but none of them have a value for C when doing an indefinate integral.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    675
    Thanks
    32

    Re: Complex Power Series

    Quote Originally Posted by olski1 View Post
    Question:

    Let F(z) be the antiderivative of f(z)=e^(z^2)with F(0).
    What do we assume about F(0)?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Apr 2010
    Posts
    56

    Re: Complex Power Series

    Thats another thing i am unsure about. That is exactly the question given. If its a typo are you able to infer what they really meant to say, because i am unsure.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Complex Power Series
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: March 22nd 2010, 08:56 PM
  2. Complex Power Series
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 10th 2010, 03:27 AM
  3. Power series (Complex)
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 25th 2008, 03:23 PM
  4. complex power series
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 23rd 2008, 11:10 AM
  5. Complex power series help
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 30th 2008, 06:27 AM

Search Tags


/mathhelpforum @mathhelpforum