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Frullani type integral

by mhill
Tags: frullani, integral, type
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mhill
#1
May11-08, 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]
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HallsofIvy
#2
May11-08, 10:51 AM
Math
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PF Gold
P: 39,534
Did you mean to say
[tex]G(t)= \int dt g(t) [/tex]
in your last line?
mhill
#3
May11-08, 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

HallsofIvy
#4
May12-08, 05:45 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,534
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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