Re: Integrate[E^x/Cos[x]]
- To: mathgroup at smc.vnet.net
- Subject: [mg38045] Re: Integrate[E^x/Cos[x]]
- From: "David W. Cantrell" <DWCantrell at sigmaxi.org>
- Date: Tue, 26 Nov 2002 00:51:15 -0500 (EST)
- References: <arsip9$een$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Peter Breitfeld <phbrf at t-online.de> wrote:
> I want to evaluate the integral:
>
> f[u_] = Integrate[E^x/Cos[x], {x, 0, u},
> Assumptions -> {x \[Element] Reals, u < Pi/2}]
>
> Mathematica 4.2 on Linux returns
>
> (1 - I)*E^((1 + I)*u)*Hypergeometric2F1[
> 1/2 - I/2, 1, 3/2 - I/2, -E^(2*I*u)] +
> (1/2)*((-I)*PolyGamma[0, 1/4 - I/4] +
> I*PolyGamma[0, 3/4 - I/4])
>
> so far so good. This integral should be real-valued, at least in the
> Range 0<=u<Pi/2.
Yes, and Mathematica's result is real-valued.
> And that is my problem. How can I get a real result
> with no imaginary I's. Neither FunctionExpand or FullSimplify did it.
If you mean that you want the result expressed without the imaginary I
appearing anywhere in it, please realize that such might be impossible.
Whether it's impossible in this particular case or not, I don't know.
Think, for example, of expressing the real roots of a cubic polynomial
in terms of nothing more sophisticated than radicals. In the so-called
_casus irreducibilis_, it is well known that, even though the roots are
real, they must, in general, be expressed using the imaginary I. Your
integral result might be similar, requiring complex numbers for its
expression in Mathematica, despite the fact that it is real.
David
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