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 $
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 ?
——
To see where “ this is coming from “ ,
Notice
$$int_0^{2 pi} ln(sin(x) + frac{5}{4}) dx = 0 $$
And look at these :
Why is $sup f_- (n) inf f_+ (m) = frac{5}{4} $?
Why is $inf g sup g = frac{9}{16} $?
—-
calculus limits proof-writing trigonometric-integrals
asked Nov 25 '18 at 22:27
mick
5,07922064
add a comment |
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 ?
——
To see where “ this is coming from “ ,
Notice
$$int_0^{2 pi} ln(sin(x) + frac{5}{4}) dx = 0 $$
And look at these :
Why is $sup f_- (n) inf f_+ (m) = frac{5}{4} $?
Why is $inf g sup g = frac{9}{16} $?
—-
calculus limits proof-writing trigonometric-integrals
asked Nov 25 '18 at 22:27
mick
5,07922064
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
add a comment |
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 ?
——
To see where “ this is coming from “ ,
Notice
$$int_0^{2 pi} ln(sin(x) + frac{5}{4}) dx = 0 $$
And look at these :
Why is $sup f_- (n) inf f_+ (m) = frac{5}{4} $?
Why is $inf g sup g = frac{9}{16} $?
—-
calculus limits proof-writing trigonometric-integrals
asked Nov 25 '18 at 22:27
mick
5,07922064
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 ?
——
To see where “ this is coming from “ ,
Notice
$$int_0^{2 pi} ln(sin(x) + frac{5}{4}) dx = 0 $$
And look at these :
Why is $sup f_- (n) inf f_+ (m) = frac{5}{4} $?
Why is $inf g sup g = frac{9}{16} $?
—-
calculus limits proof-writing trigonometric-integrals
calculus limits proof-writing trigonometric-integrals
asked Nov 25 '18 at 22:27
mick
5,07922064
asked Nov 25 '18 at 22:27
mick
5,07922064
edited Nov 25 '18 at 22:33
asked Nov 25 '18 at 22:27
mick
5,07922064
asked Nov 25 '18 at 22:27
mick
5,07922064
asked Nov 25 '18 at 22:27
mick
5,07922064
5,07922064
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
add a comment |
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
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
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
I think the conjecture is false actually but hard to show false ?
– mick
Nov 27 '18 at 20:21
add a comment |
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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