How are you getting from
to
?
Shouldn't it be
?
No suggestions on how to proceed with this, but
You know me well sir. I already was there deriving them, now most of them I can see what to do. But one is alluding me . Now I haven't given it much time butHey mathstud, you know what might be fun?. Go to the link I posted above and actually derive the solutions they give for the integrals. You have probably done that already.
Some of them are probably very challenging.
Ok my first move was to do the obvious double integral
So we have
Now rewriting this as (this is so people who aren't super accustomed to double integrals can see what I am doing)
Where
From here it is obvious that this region is a rectangle, so Fubini's Theorem applies
So
Now this inner integral is bothering me, I cannot seem to get it in the eight minutes I have tried
parts-no deal
Letting -no go
Any suggestions?
I also tried some crazy thing where I let
And found that the integral I want is a constant difference from
<-- with an f. Now I feel stupid for copying it down wrong
Well
By Integration by Parts twice then adding to both sides.
So now as x goes to infinity obviously the overpowers and it goes to zero, and when the above equals
So we have that
So now we must perform
Rewriting this as
And letting
So we have
Back subbing we get
Seeing that I did a-b opposed to b-a this is what they had
My 89 doesn't give a solution and Mathematica tells me this:
Why don't you try to solve this using infinite series?
You know the power series expansions better than I do. Experiment with this and let me know how you do. I'll try to remember what I can to see I can evaluate the integral...
Wait...can this be related to the zeta function, by any chance [I'm probably wrong...just throwing ideas out there]?
--Chris
The other one
Ok once again seeing that
We have that is equivalent to
Once again seeing the region of integration is a rectangle we may switch the integration order without changing the limits by Fubini's Theorem
Now once again by double integration by parts and algebra
Now
and evaluated at zero is
so
So using
We see that
Now for
If we let then
So we have
So back subbing we get
So
Now putting together we get