Find .
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Posts Tagged ‘Double Integral’
Monday Math 148
December 27, 2010Monday Math 142
November 8, 2010Find :
1). Using an infinite series and the Riemann zeta function
2). Using neither infinite series nor polylogarithms
Solutions:
Monday Math 79: A Challenge
July 6, 2009A challenge to my readers:
Find the exact value of the double integral .
Monday Math 55
January 19, 2009How might one perform the unit-square integral
?
First, let us swap the order of integration:
.
Next, we approach the inner integral by performing the u-substitution
→
The limits become y=0 → u=∞, y=1 → u=-ln(x).
So, with this substitution, we find , and
for 0<x<1, ln(x)<0, so -ln(x)>0; -ln(x)→∞ as x→0, and -ln(1)=0, so the integration region can be seen as between and x=1, u>0. So, reversing the order of the double integral,
We found previously that , and the gamma function is defined as ; this means
This is called Hadjicostas’s formula, and holds for s any complex number with .
Note that as the pole of ζ(s) at s=1 is of order one: the Laurent series is of the form ,
where the are called the Stieltjes constants, and , the Euler-Mascheroni constant; the singularity in the term is thus removable, with
Thus we have
.
Similarly, with the related integral
,
we can use the same subtitution and change of order:
.
What if we try instead try the integral
?
Performing the same u-substitution:
.
Reversing the order of the double integration, and performing the inner integral,
We found previously that , so then
(for )
and, in analogy to I(s), we can also find that.
Monday Math 28
July 14, 2008Let us consider a function of two real variables α and β; such that is defined and continuous for all α,β>0; and which obeys, for α≠β,
.
Does such a function exist, and if so, what is it?
Note that this exercise has three parts: First, to show that the integrand is valid over the infinite interval (0,∞); second, that the discontinuity in on the line α=β is removable; and third, to evaluate the integral, and thus find the function.
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