# Talk:Dirichlet integral

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Field:  Analysis

The proof is very confusing, there might be minor mistakes. Also, I don't think you need to bring in the second variable (beta). On the Bosnian version of the page the proof is in one variable and it is much clearer.

  The beta variable just makes it more general.


There is a step where the imaginary part of a complex exponential is made to include the whole integral, is that justified? —Preceding unsigned comment added by 188.62.241.118 (talk) 20:04, 17 April 2010 (UTC)

## Using Complex Analysis

Can this be proven using Complex Analysis? --Hirak 99 (talk) 09:30, 12 August 2010 (UTC)

## Complex Integration

The method here seems unnecessarily difficult. In particular I think it is preferable not to have to refer to another page (Sokhotski-Plemelj theorem) - especially when that theorem is not even proved on that page!

The obvious alternative would be to integrate ${\displaystyle f(z)={\frac {e^{iz}}{z}}}$ around a composite path parameterised by ε → 0 and R → ∞, consisting of the real line segments [-R,-ε] and [ε,R], the semicircle diameter R centred about zero in the upper half-plane in the positive direction, and the semicircle diameter ε centered about zero in the upper half-plane in the negative direction. The whole integral is zero, the integral around the large semicircle tends to zero and the integral around the small semicircle tends to -iπ (via Taylor expansion of the exponent function), so the integral on the real line segments of sin(x)/x is π. Finally the function is even, so the integral over the positive real line is π/2.

It might also be worth justifying the statement that the integral around the large semicircle tends to zero.

I think that would make the page accessible to the average second-year maths undergraduate.

BobHatt (talk) 19:04, 23 January 2013 (UTC)

## Another Proof?

Using the function ${\displaystyle f(a)=\int _{\infty }^{\infty }{\frac {sin(ax)}{x}}dx}$ and differentiating wrt to a, under the integral sign, the result can be shown very easily(ultimately using the elementary property of delta functions). But I'm not sure if after differentiating, it is correct to write ${\displaystyle f'(a)=\int _{\infty }^{\infty }cos(ax)dx=2\pi \delta (a)}$ . Should that go into this page? Aritrop (talk) 03:22, 25 April 2015 (UTC)

It's just the g(1) value of the bottom of section 1.2. Cuzkatzimhut (talk) 20:21, 22 January 2016 (UTC)