Most of us know of the fact that . Euler has calculated the exact this sum in his time. Eulers solution to Basel problem. Today we are going to prove it using calculus, first by showing that a certain double integral of a function over a region is equivalent to when expressed as its infinite series , then by directly finding the exact value of double integral using transformations . Since both forms are equivalent we can prove the sum.
Consider the double integral over the rectangular region bounded by points .
Now we express this function as an infinite series , so the integral becomes
Now we are going to find the definite integral of this function in another way:
First we transform the following the region to another space which we will call space using the following equations . This transformation is linear therefore the transformed region will also be a rectangular region. The only difference is that it is rotated . The jacobian for the above transformation is
The points bounding the transformed regions :
Now we rewrite the double integral:
from the transformation we can split this integral into 2 as following
This is because the region up to point is enclosed within lines and and the region from this point to is enclosed within lines and
Now we can iterate through this double integral first w.r.t v and then u
u is independent to v.
Therefore it is equal to
Now next iteration
Limits in terms of theta are: to to
Now we do the second part of the split integral :
Again we iterate through v first and then u
Anti derivative for integration w.r.t v is same ,only the limits change.
Again using the fact that
now the next iteration:
Now we know
Now add this to which is equal to
going back a few steps :
going back to our integral :
Now to express limits in terms of m :
from to is equal to limit from to
So limits in terms of is equal to to
in terms of m :
Now finally adding this to the result from our first split integral ,