Another quick contour integral from the integralbot
In this blog entry we solve another integral from @integralbot via contour integration. Click here for the proof.
Posts
QUICK CONTOUR INTEGRAL FROM @integralbot
In today´s post we will prove the following result by contour integration Click here to read the post.
A log trig integral \int_0^{\pi/2}\ln\left(1+a\sin^2 x \right) \,dx
In this post we will prove the following result Click here for the proof We have relied on the previous established result Proof here .
log log integral inverse sech
In this entry we will prove the following result Click here to read it. Some previous established results were used, i.e.: Lerch´s formula Click here for the proof. Relation between Bernoulli´s polynomials and Hurwitz zeta function click here for the proof (equations 8.22 and 8.30)
Series involving reciprocal of the central binomial coefficient squared
Today we will prove this infinite series which involves the reciprocal of the central binomial coefficient squared, namely Click here for the proof.
Central Binomial representation for zeta(2)
In this post we will prove the following nice series representation for : Click here for the proof. We used the previous established result Click here for the proof.
Central Binomial coefficient series and zeta(2)
In this entry we will prove the following result that appears in this Twitter post Click here for the proof. In the proof we used the previous result proved here .
Series involving central binomial coefficient squared
In this blog entry we will evaluate the beautiful series below involving the square of the central binomial coefficient Click here to see the proof We used the previous result proved here
Central Binomial coefficient generating function
Today we will prove the following generating function that appears in this Twitter post Click here for the proof.
CHALLENGING INTEGRAL INTEGRALBOT
In this post we will prove the following integral Click here to see the proof. We will need the following results previously established proved here , and proved here
HYPERBOLIC INTEGRAL WITH GLAISHER CONSTANT
Today we will prove the following result seen in this Twitter post : First we need a Lemma: Lemma 1: (1) Proof: Corollary: If we let in (1) we obtain (2) Now, consider the following integral (3) Proof: Where in (*) we used the following result proved here Now if we differentiate (3) with respect to s and let we get where we used that (see here ) and
ELLIPTIC INTEGRAL HYPERBOLIC FUNCTIONS
Today we will prove the following result that appears in this Twitter post Click here to see the proof.
INTEGRALBOT BETA-DIGAMMA INTEGRAL
Today we will show the following result that appears in this twitter post We will start by proving two lemmas involving the Beta and Digamma functions. Lemma 1: (1) Recall the Beta function (2) If we differentiate (2) w.r. to s we obtain Lemma 2: (3) Recall the Beta function (4) If we differentiate (4) w.r. to a we obtain Now let´s evaluate the integral: For the proof of the special values of the Digamma function see this post . Plugging the values of J and K back in the original integral we obtain:
@integralsbot \int_0^\infty \left(\sqrt{1+x^4}-x^2\right) \,dx=\frac{\Gamma^2\left( \frac14\right)}{6 \sqrt{\pi}}
Today we will show the following result that appears in this post from @integralsbot Let Then: And We then get: By the reflection formula Letting we obtain that By the functional equation of the Gamma function We obtain for instance that