Monday Math 76

Find \int_0^{\pi/2}\sqrt{\tan\theta}\,d\theta.


First, consider the beta function, and note that
\begin{array}{rcl}\beta(x,(1-x))&=&\frac{\Gamma(x)\Gamma(1-x)}{\Gamma(x+(1-x))}\\&=&\frac{\Gamma(x)\Gamma(1-x)}{\Gamma(1)}\\&=&\Gamma(x)\Gamma(1-x)\end{array}
and, using the Euler reflection formula,
\beta(x,(1-x))=\frac{\pi}{\sin(\pi{x})}.

How does this relate to our integral? One may recall that
\beta(x,y)=2\int_{0}^{\frac{\pi}{2}}(\cos\theta)^{2x-1}(\sin\theta)^{2y-1}\,d\theta.

Now, \sqrt{\tan\theta}=\sqrt{\frac{\sin\theta}{\cos\theta}}=(\cos\theta)^{-\frac{1}{2}}(\sin\theta)^{\frac{1}{2}}.
Thus,
\begin{array}{rcl}\int_0^{\pi/2}\sqrt{\tan\theta}\,d\theta&=&\int_0^{\pi/2}(\cos\theta)^{-\frac{1}{2}}(\sin\theta)^{\frac{1}{2}}\,d\theta\\&=&\frac{1}{2}\beta\left(\frac{1}{4},\frac{3}{4}\right)\\&=&\frac{1}{2}\frac{\pi}{\sin\frac{\pi}{4}}\\&=&\frac{\pi}{\sqrt2}\end{array}.

In fact,
\begin{array}{rcl}\int_0^{\pi/2}\sqrt[n]{\tan\theta}\,d\theta&=&\int_0^{\pi/2}(\tan\theta)^{\frac{1}{n}}\,d\theta\\&=&\int_0^{\pi/2}(\cos\theta)^{-\frac{1}{n}}(\sin\theta)^{\frac{1}{n}}\,d\theta\\&=&\frac{1}{2}\beta\left(\frac{1}{2}\left(1-\frac{1}{n}\right),\frac{1}{2}\left(1+\frac{1}{n}\right)\right)\\&=&\frac{1}{2}\frac{\pi}{\sin\left(\frac{\pi}{2}\left(1-\frac{1}{n}\right)\right)}\\&=&\frac{\pi}{2\cos\left(\frac{\pi}{2n}\right)}\end{array}>.

(See here for a similar use of the beta function and Euler reflection formula.)

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2 Responses to “Monday Math 76”

  1. martin flaxman Says:

    Hi I’m an amateur mathematician. I’ve been trying for about year to get the lowdown on the beta function.

    I need to know why and how Euler derived the function, its significance, and how and when it is used.

    Any clue will be appreciated.

    Thanks, Martin Flaxman

    • twistedone151 Says:

      Here is a short biography of Euler, which mentions that he introduced the beta and gamma functions, as integrals, in 1729, and that they were named by later mathematicians. In particular:

      He also studied beta and gamma functions, which he had introduced in 1729. Legendre called these ‘Eulerian integrals of the first and second kind’ respectively while they were given the names beta function and gamma function by Binet and Gauss respectively.

      One use of the beta function is as the normalization factor in the beta distribution. It also arises in the stochastic urn process known as a preferential attachment process.
      One might also read about physicist Gabriele Veneziano, who, in applying the beta function to the scattering amplitudes of particles interacting via the strong force, led to the creation of string theory.
      I hope these give you places to start

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