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Frullani type integral 
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#1
May1108, 08:15 AM

P: 193

under what condition the does the equality hold ?
[tex] \int_{0}^{\infty} dt \frac{ f(at)f(bt)}{g(t)}= (G(b)G(a))(f(0)f(\infty)) [/tex] and [tex] \int dt g(t) [/tex] 


#2
May1108, 10:51 AM

Math
Emeritus
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Thanks
PF Gold
P: 39,569

Did you mean to say
[tex]G(t)= \int dt g(t) [/tex] in your last line? 


#3
May1108, 05:03 PM

P: 193

yes i meant the integral of g(t) ,i have been looking for generalizations of this integral but without any luck , under that condition can define g(t) and f(at) f(bt) so the integral above converge ?? thank



#4
May1208, 05:45 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,569

Frullani type integral
The generalized mean value theorem says that if f and g are differentiable on (a, b) and continuous on [a, b], then there exist c in (a, b) such that
[tex]\frac{f(b) f(a)}{g(b) g(a)}= \frac{f'(c)}{g'(c)}[/tex] That looks like an integral version to me. 


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