Special Functions

From Exampleproblems

viertroc These definitions are useful.

Gamma function $\Gamma(z) = \int_0^\infty e^{-t} t^{z-1} \,dt\,$

Bessel function (First Kind) $J_p(z) = \sum_{k=0}^\infty \frac{(-1)^k\left(\frac{z}{2}\right)^{2k+p}}{k!\Gamma(k+p+1)}\,$

Hypergeometric function ${}_pF_q(a_1,...,a_p;c_1,...,c_q;z) = \sum_{n=0}^\infty \frac{(a_1)_n\cdot\cdot\cdot(a_p)_n}{(c_1)_n\cdot\cdot\cdot(c_q)_n}\frac{z^n}{n!}\,$

1. solution $\int_0^\infty x^3 e^{-x}dx\,$

2. solution $\int_0^\infty x^6 e^{-2x}dx\,$

3. solution Show that $\Gamma(n+1) = n \Gamma(n), n>0\,$

4. solution $\frac{\Gamma(6)}{2\Gamma(3)}\,$

5. solution $\frac{\Gamma(5/2)}{\Gamma(1/2)}\,$

6. solution $\frac{\Gamma(3)\Gamma(2.5)}{\Gamma(5.5)}\,$

7. solution $\frac{6\Gamma(8/3)}{5\Gamma(2/3)}\,$

8. solution $\Gamma(1/2)\,$

9. solution $\int_0^\infty \sqrt{y} e^{-y^3} dy\,$

10. solution $\int_0^\infty 3^{-4z^2}dz\,$

11. solution $\int_0^1 \frac{1}{\sqrt{-\ln x}} dx\,$

12. solution $\int_0^\infty x^m e^{-ax^n} dx\,$

13. solution $\sum_{k=0}^n {n \choose k}^2\,$

14. solution $\int_0^\infty e^{-st}t^{x-1}\,dt,\,\,\,s>0\,$

15. solution $\int_0^\infty x^4 e^{-x^3} \,dx\,$

16. solution $\int_0^\infty e^{-pt} \sqrt{t}\,dt\,$