MHB Integrals: Show $\alpha$ Not Int Multiple of $\pi$ and $s, \lambda >0$

AI Thread Summary
For $\alpha$ not an integer multiple of $\pi$, the integral from $-\frac{\pi}{4}$ to $\frac{\pi}{4}$ of $\left( \frac{\cos x + \sin x}{\cos x - \sin x} \right)^{\cos \alpha}$ equals $\frac{\pi}{2 \sin \left(\pi \cos^{2} \frac{\alpha}{2} \right)}$. A correction was made regarding the second integral, confirming that it should involve $x^{s-1}$ instead of $t^{s-1}$. The second integral evaluates to $\frac{\Gamma(s) \cos (\alpha s)}{\lambda^{s}}$ for $s,\lambda >0$ and $0 \le \alpha < \frac{\pi}{2}$. An alternative solution using contour integration was also discussed, emphasizing the use of Jordan's inequality. The discussion highlights the importance of proper variable notation and the application of advanced integration techniques.
polygamma
Messages
227
Reaction score
0
1) Show that for $\alpha$ not an integer multiple of $\pi$, $\displaystyle \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \Big( \frac{\cos x + \sin x}{\cos x - \sin x} \Big)^{\cos \alpha} \ dx = \frac{\pi}{2 \sin \left(\pi \cos^{2} \frac{\alpha}{2} \right)} $.2) Show that for $s,\lambda >0$ and $0 \le \alpha < \frac{\pi}{2}$, $\displaystyle \int_{0}^{\infty} x^{s-1} e^{-\lambda x \cos \alpha} \cos(\lambda x \sin \alpha) \ dx = \frac{\Gamma(s) \cos (\alpha s)}{\lambda^{s}}$.
 
Last edited:
Mathematics news on Phys.org
I think there is a mistake in the second question $$t^{s-1}$$ is completely independent of the integral so it can be taken out of the integration .

Maybe , you meant instead $$x^{s-1}$$
 
ZaidAlyafey said:
I think there is a mistake in the second question $$t^{s-1}$$ is completely independent of the integral so it can be taken out of the integration .

Maybe , you meant instead $$x^{s-1}$$

Yes. That's what I meant. Thanks.
 
Random Variable said:
1) Show that for $\alpha$ not an integer multiple of $\pi$, $\displaystyle \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \Big( \frac{\cos x + \sin x}{\cos x - \sin x} \Big)^{\cos \alpha} \ dx = \frac{\pi}{2 \sin \left(\pi \cos^{2} \frac{\alpha}{2} \right)} $.
Cool problem : $$ \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \Big( \frac{\cos x + \sin x}{\cos x - \sin x} \Big)^{\cos \alpha} \ dx$$

$$ \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \left( \frac{1+\sin(2x)}{\cos(2x)} \right) ^{\cos \alpha} \, dx$$

Now we use the change of variable : $$t=\frac{\pi}{4}-x$$

$$\int^{\frac{\pi}{2}}_0 \left(\frac{1+\cos 2t }{\sin 2t } \right)^{\cos \alpha }\, dt$$

$$ \int ^{\frac{\pi}{2}}_0 (\cos t )^{\cos \alpha } \cdot (\sin t)^{-\cos \alpha }dt$$

Now using the beta identity we get :

$$\frac{1}{2} \text{B} \left( \frac{1+\cos \alpha }{2}, \frac{1-\cos \alpha }{2}\right) \, = \frac{1}{2} \Gamma \left(\frac{1+\cos \alpha }{2}\right) \cdot \Gamma \left(\frac{1-\cos \alpha }{2}\right)$$

Now we use the reflection formula to get :

$$\frac{\pi}{2 \sin \left( \pi \cos^2 \left( \frac{\alpha}{2} \right) \right) }$$
 
A slightly different approach is $ \displaystyle \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \Big( \frac{\cos x + \sin x}{\cos x - \sin x} \Big)^{\cos \alpha} \ dx = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \Bigg( \frac{ \sin \left(\frac{\pi}{4} +x \right)}{\sin \left(\frac{\pi}{4}-x\right)}\Bigg)^{\cos \alpha} dx $

$ \displaystyle = \int_{0}^{\frac{\pi}{2}} \Bigg( \frac{ \sin \left(\frac{\pi}{2} - u \right)}{\sin u}\Bigg)^{\cos \alpha} du = \int_{0}^{\frac{\pi}{2}} (\cos u)^{\cos \alpha} (\sin u)^{-\cos \alpha} \ du$
 
I solved the second one , should I post the solution ?
 
Random Variable said:
2) Show that for $s,\lambda >0$ and $0 \le \alpha < \frac{\pi}{2}$, $\displaystyle \int_{0}^{\infty} x^{s-1} e^{-\lambda x \cos \alpha} \cos(\lambda x \sin \alpha) \ dx = \frac{\Gamma(s) \cos (\alpha s)}{\lambda^{s}}$.
.
.
.
.
.
.
.

Here is the solution for the second one :

$$\cos(\lambda x \sin \alpha) = \frac{e^{ i\lambda x \sin \alpha}+e^{-i\lambda x \sin \alpha}}{2}$$

$$\int_{0}^{\infty} x^{s-1} e^{-(\lambda x \cos \alpha )} \left( \frac{e^{ i\lambda x \sin \alpha}+e^{-i\lambda x \sin \alpha}}{2} \right) dx $$

$$\frac{1}{2}\int_{0}^{\infty} x^{s-1} e^{-\lambda x (\cos \alpha -i\sin \alpha)}\, dx+\, \frac{1}{2}\int_{0}^{\infty} x^{s-1} e^{-\lambda x (\cos \alpha +i\sin \alpha)} \, dx$$

$$\frac{1}{2}\int_{0}^{\infty} x^{s-1} e^{- \left(\lambda \,e^{i \alpha}\,x\right)}\, dx+\, \frac{1}{2}\int_{0}^{\infty} x^{s-1} e^{- \left(\lambda \,e^{-i \alpha}\, x\right)} \, dx$$

Then we use the Laplace transform :

$$\frac{\Gamma(s)}{ 2\left(\lambda \,e^{i \alpha}\right)^s }\,+\,\frac{\Gamma(s)}{2\left(\lambda \,e^{-i \alpha}\right)^s }\,=\frac{\Gamma(s)}{\lambda^s} \left( \frac{\,e^{i \alpha \, s}+\, e^{-i \alpha \, s}}{2} \right)\, = \, \frac{\Gamma(s) \cos (\alpha s)}{\lambda^{s}}$$
 
If you have another approach, please share it ... :cool:
 
Here's an alternative solution using contour integration.

Let $f(z) = z^{s-1}e^{-\lambda z}$ and integrate around a sector in the first quadrant that forms an angle of $\alpha$ with the positive real axis. The contour needs to be indented at the origin since $z=0$ is a branch point.

The integral evaluates to zero along the big arc and the small arc in the limit. Just use Jordan's inequality.

So we have $\displaystyle \int_{0}^{\infty} f(x) dx - \int_{0}^{\infty} f(te^{ia}) \ dt =0 $

or $ \displaystyle \int_{0}^{\infty} x^{s-1} e^{-\lambda x} \ dx - e^{ias} \int_{0}^{\infty} t^{s-1} e^{-\lambda t \cos \alpha} e^{-i \lambda t \sin a} \ dt =0$

$\displaystyle \implies \int_{0}^{\infty} t^{s-1}e^{-\lambda t \cos \alpha} \Big(\cos(\lambda t \sin \alpha)- i \sin (\lambda t \sin\alpha) \Big) \ dt = e^{-ias} \int_{0}^{\infty}x^{s-1} e^{-\lambda x} \ dx = e^{-ias} \frac{\Gamma(s)}{\lambda^{s}}$

$ \displaystyle = \Big(\cos (as) - i \sin (as) \Big) \frac{\Gamma(s)}{\lambda^{s}}$

Now just equate the real parts on both sides of the equation.
 
Back
Top