In this previous post, I claimed that , where is the zeta function There are several proofs for . In this post, I share one from Reference 1 that uses Fourier series. Reference 1 details how one can evaluate for Continue reading
The zeta function is defined as In Reference 1, I learnt of a pretty simple double integral which evaluates to , which in turn is well-known to be equal to . Here is the result: This is not too hard Continue reading
The zeta function is defined as and the gamma function is defined as The integral representation of the zeta function refers to the following relationship: Proposition. For complex with , we have The following hand-wavy proof below is due to Continue reading
Nahin s In Pursuit of Zeta-3 (Reference 1) is full of many interesting theorems and results connecting different important functions in math. In this post, I share a connection between Euler s constant and the zeta function that he presents in the Continue reading
The Gaussian integral, sometimes known as the probability integral, is the result In In Pursuit of Zeta-3 (Reference 1), Nahin presents and proves a generalization of the Gaussian integral: for any fixed constants , (We obtain the Gaussian integral by Continue reading
I learnt of the following statement from John Cook s blog: Let and denote the golden ratio and its inverse respectively: If a function is such that Then we must have for all . It s straightforward to verify that satisfies all Continue reading
Evaluate Solution: For any finite , The first product is easy to reduce: If you write our the first few terms of the second product, it s not hard to come to the following in order to get telescoping cancellation: Putting Continue reading
The napkin ring problem concerns finding the volume of a solid after a hole has been drilled into a sphere: Take a sphere of radius r and drill out a hole along a diameter so the remaining shape has height Continue reading
The Ford circles, introduced by L. R. Ford in his 1938 paper (Reference 1), are a family of circles that are all tangent to the -axis at rational points. For each rational number with , the Ford circle is defined Continue reading
We define the Itô integral as where the partition gets increasingly fine as , and is a standard Wiener process (Brownian motion). The Itô integral is defined for stochastic processes that satisfy the following two assumptions (let s call them the Continue reading