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Topic: [HM] Integral of sinx /x
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Hans Fischer

Posts: 15
Registered: 12/3/04
[HM] Integral of sinx /x
Posted: Oct 27, 2005 5:52 AM
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Dear list members,

a quite elementary proof of the fact, that the integral of sin x/x
between 0 and infinity is equal to pi/2 runs as follows:

1.) Show, that the integral (conditionally) converges.

2.) Because of (sin (2n+1) x)/sin x = 1 + 2cos2x + 2 cos 4x + ....+ 2
cos 2nx (n a natural number) follows, that
the integral of (sin (2n+1) x)/sin x between 0 and pi/2 is equal to
pi/2 for all natural n.

3.) Show, that the function f(x) = 1/x - 1/sin x has a continuous
derivative for all x in [0; pi/2].

4.) Show by partial integration, that the integral of (sin (2n+1)x)(1/x
- 1/sin x) between 0 and pi/2 tends to 0 if n tends to infinity.

Then we are done, because the integral of sin x/x between 0 and infinity
is equal to the limit of the integral of sin (2n+1) x/x between 0 and
pi/2 as n tends to infinity.

In principle, the idea of this proof is implicitly contained in
Dirichlet's articles on the convergence of Fourier series. The earliest
complete version of this proof, which I can find, however, is in
Courant's "Differential- und Integralrechnung", Vol. I, 2nd(!) ed.,
1930, pp. 371/72.

Does anybody know an earlier source for this particular proof?

Thank you in advance and best wishes

Hans Fischer

Hans Fischer
Katholische Universitaet Eichstaett
Mathematisch-Geographische Fakultaet
D-85071 Eichstaett
Tel. +49 8421 93 1256
Tel. priv. +49 8421 80919
Fax: +49 8421 2256

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