About $ f(w,L) = \int_1^w \int_0^{2 L \pi} \frac{ \ln(\frac{\sin(x) +\ sin(vx)}{2} + \frac{5}{4})}{L(w - 1)} dx dv $

Asked Nov 25 '18 at 22:27

Active Nov 25 '18 at 22:33

Viewed 69 times

Consider

$$ f(w,L) = \int_1^w \int_0^{2 L \pi} \frac{ \ln(\frac{sin(x) + sin(vx)}{2} + \frac{5}{4})}{L(w - 1)} dx dv $$

For real $w > 1 $ and integer $ L > 1$

Conjecture :

$$ \lim_{L \to \infty} f(w + 1, L) - f(w,L) = 0. $$

How to decide if this is true ?

Perhaps differentiation under the integral sign is the best method ?

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To see where “ this is coming from “ ,

Notice

$$\int_0^{2 \pi} \ln(sin(x) + \frac{5}{4}) dx = 0 $$

And look at these :

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edited Nov 25 '18 at 22:33

asked Nov 25 '18 at 22:27

mick

Although the Integral looks complicated it might be expressible in special functions ? I was thinking about hypergeometric. If that would be simpler or even helpful is another matter ... – mick Nov 25 '18 at 22:48
I think the conjecture is false actually but hard to show false ? – mick Nov 27 '18 at 20:21

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