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From: lambert@boring.uucp (Lambert Meertens)
Newsgroups: net.math
Subject: Re: value of an integral
Message-ID: <6806@boring.UUCP>
Date: Mon, 3-Mar-86 09:48:06 EST
Article-I.D.: boring.6806
Posted: Mon Mar  3 09:48:06 1986
Date-Received: Wed, 5-Mar-86 05:49:38 EST
References: <3@bgsuvax.UUCP>
Reply-To: lambert@boring.UUCP (Lambert Meertens)
Organization: CWI, Amsterdam
Lines: 31
Apparently-To: rnews@mcvax

> [...] does anyone out there know if the indefinite integral of e**x*sec(x)
> is expressible in terms of elementary functions?

The indefinite integral can be written as a Fourier-like series:

        x  cos x + sin x   cos 3x + 3 sin 3x   cos 5x + 5 sin 5x
  C + 2e  (------------- - ----------------- + ----------------- - ... ) .
	     1  +  1^2        1   +   3^2         1   +   5^2

This can be found by using the formal series

                ix   3ix    5ix
    sec x = 2 (e   -e    + e    + ... ) .

The integration constant C jumps at the poles of the integrand.
I don't see how to rewrite the sum as a finite closed expression, although it
does not look beyond hope; in particular, it is reminiscent of the expansions

              2m             1      cos x     cos 2x    cos 3x
    cosh mx = -- sinh m.pi (---- - ------- + ------- - ------- + ... )
              pi            2m^2   m^2+1^2   m^2+2^2   m^2+3^2

               2             sin x    2 sin 2x   3 sin 3x
    sinh mx = -- sinh m.pi (------- + -------- - -------- + ... ) .
              pi            m^2+1^2    m^2+2^2    m^2+3^2

-- 

     Lambert Meertens
     ...!{seismo,okstate,garfield,decvax,philabs}!lambert@mcvax.UUCP
     CWI (Centre for Mathematics and Computer Science), Amsterdam