Evaluation of Integrals Using Special Functions - One Line Questions
1.
What is the value of $\Gamma(n)$ for a positive integer n? —
(n-1)!
2.
The integral $\int_0^x e^{-t^2/2} dt$ is related to the error function by: —
$\frac{1}{2} erf(x/\sqrt{2})$
3.
The integral $\int_0^\infty t^{m-1} e^{-at} dt$ can be evaluated using the Gamma function and is equal to: —
$\Gamma(m) / a^m$
4.
The integral $\int_0^1 x^{m-1} (1-x)^{n-1} dx$ is known as the Beta function and is related to the Gamma function by which formula? —
$\Gamma(m) \Gamma(n) / \Gamma(m+n)$
5.
Which identity relates the Gamma function to the factorial for non-negative integers? —
$\Gamma(n+1) = n!$
6.
The first kind of complete elliptic integral is denoted by $K(k)$ and is given by which integral? —
$\int_0^{\pi/2} \frac{d\theta}{\sqrt{1-k^2 \sin^2 \theta}}$
7.
The second kind of complete elliptic integral is denoted by $E(k)$ and is given by which integral? —
$\int_0^{\pi/2} \sqrt{1-k^2 \sin^2 \theta} d\theta$
8.
Which integral is defined as the first kind of complete elliptic integral? —
$\int_0^{\pi/2} \frac{d\theta}{\sqrt{1-k^2 \sin^2 \theta}}$
9.
Which integral represents the second kind of complete elliptic integral? —
$\int_0^{\pi/2} \sqrt{1-k^2 \sin^2 \theta} d\theta$
10.
Which integral is defined by the Beta function $B(m, n)$? —
$\int_0^1 x^{m-1} (1-x)^{n-1} dx$
11.
Which integral is defined by the first kind of incomplete elliptic integral, $F(\phi, k)$? —
$\int_0^\phi \frac{d\theta}{\sqrt{1-k^2 \sin^2 \theta}}$
12.
Which integral representation is characteristic of the second kind of incomplete elliptic integral, $E(\phi, k)$? —
$\int_0^\phi \sqrt{1-k^2 \sin^2 \theta} d\theta$
13.
Which of the following is NOT a standard form for the Beta function? —
$\int_0^1 \frac{x^{m-1}}{1+x} dx$
14.
Which integral is defined by the Beta function $B(m, n)$? —
$\int_0^1 x^{m-1} (1-x)^{n-1} dx$
15.
What is the value of $\Gamma(1/2)$ related to? —
$\sqrt{\pi}$
16.
The integral $\int_0^\infty e^{-x^2} dx$ is equal to: —
$\sqrt{\pi}/2$
17.
What is the value of the Gamma function $\Gamma(1/2)$? —
$\sqrt{\pi}$
18.
The integral $\int_{-\infty}^\infty e^{-x^2} dx$ is a famous Gaussian integral, and its value is: —
$\sqrt{\pi}$
19.
The integral of the form $\int R(\sin x, \cos x) dx$, where R is a rational function, can often be reduced to an integral of a rational function using a substitution involving which function? —
$\tan(x/2)$
20.
The Gamma function satisfies $\Gamma(z+1) = z\Gamma(z)$. If $z=3$, then $\Gamma(4)$ is equal to: —
$3!$
21.
The integral $\int_0^\pi \sin^n x dx$ can be evaluated using the Beta function when n is an integer. What is the relationship? —
$B((n+1)/2, 1/2) / 2$
22.
The integral $\int_0^1 x^a dx$ can be evaluated using the Beta function by setting $n=1$. The result is: —
$B(a+1, 1) = 1/(a+1)$
23.
The integral $\int_0^1 x^a dx$ can be seen as a specific case of the Beta function $B(m, n)$ where $m-1=a$ and $n-1=0$. This gives: —
$B(a+1, 1) = 1/(a+1)$
24.
The integral $\int_0^1 x^a (1-x)^b dx$ evaluates to: —
$B(a+1, b+1)$
25.
The integral $\int_0^1 x^a (1-x)^b dx$ is equal to: —
$B(a+1, b+1)$
26.
The integral $\int_0^\infty \frac{t^{m-1}}{1+t} dt$ is related to the Beta function and equals: —
$B(m, 1-m)$ for $0 < m < 1$
27.
The Beta function $B(m, n)$ is symmetric, meaning: —
$B(m, n) = B(n, m)$
28.
The integral $\int_0^1 x^{m-1} (1-x^2)^{n-1} dx$ can be evaluated using the Beta function. The substitution $u=x^2$ leads to: —
$B(m/2, n)/2$
29.
The integral $\int_0^1 x^{m-1} (1-x^n)^p dx$ can be evaluated using the Beta function after the substitution $u=x^n$. This results in: —
$B(m/n, p+1)/n$
30.
Which form of the Beta function is useful for evaluating integrals of the type $\int_0^{\pi/2} \sin^p x \cos^q x dx$? —
$B((p+1)/2, (q+1)/2)$
31.
The Fresnel integrals, S(x) and C(x), are defined by integrals involving which function? —
$\sin(t^2)$ and $\cos(t^2)$
32.
The integral $\int_0^1 x^a (1-x^b)^c dx$ can be evaluated using the Beta function after a suitable substitution. What is the general form of the substitution? —
$x = u^{1/b}$
33.
The error function, erf(x), is defined as $ \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt $. What is its value at x=0? —
0
34.
What is the value of $\Gamma(1)$? —
1
35.
The Gamma function $\Gamma(z)$ is defined for which values of z? —
All complex numbers except non-positive integers
36.
Which special function is defined by the integral $\int_0^\infty x^n e^{-ax^2} dx$ for suitable n and a? —
Gamma Function
37.
The integral $\int_0^\infty x^m e^{-x} dx$ defines which special function? —
Gamma Function
38.
The integral $\int_0^\infty t^k e^{-t^2} dt$ is evaluated using which special function? —
Gamma Function
39.
The integral $\int_0^\infty t^{n-1} e^{-t} dt$ is the definition of: —
Gamma Function $\Gamma(n)$
40.
The integral $\int_0^x \sin(t^2) dt$ defines which special function? —
Fresnel Sine Integral S(x)
41.
Which special function is commonly used to evaluate integrals involving trigonometric functions, particularly in the context of Fourier series? —
Elliptic Integral
42.
The integral $\int_{-\infty}^\infty e^{-ax^2} dx$ for $a>0$ is related to which special function? —
Gamma Function
43.
The integral $\int_0^x e^{-t^2} dt$ is directly related to which special function? —
Error Function
44.
Which special function is defined by the integral $\int_0^1 x^m (1-x)^n dx$? —
Beta Function
45.
The integral $\int_0^1 \frac{dx}{\sqrt{1-x^n}}$ for $n>2$ is related to which class of special functions? —
Elliptic Integrals
46.
Which type of integral involves the square root of a polynomial of degree 3 or 4 under the integral sign, and is fundamental in the study of elliptic curves? —
Elliptic Integrals
47.
Which property of the Gamma function, $\Gamma(z+1) = z\Gamma(z)$, is crucial for its relation to factorials? —
Recurrence Relation
48.
Bessel functions are solutions to Bessel's differential equation. Their integral representations often involve which type of functions? —
Trigonometric and exponential functions
49.
Which property of the Gamma function is stated as $\Gamma(z) \Gamma(1-z) = \frac{\pi}{\sin(\pi z)}$? —
Reflection Formula
50.
The integral $\int_0^1 \frac{x^{m-1}}{1+x} dx$ can be expressed in terms of the Beta function when $m$ is related to another parameter. This form is less common than the standard integral. —
It equals $B(m, 1-m)$ for $0 < m < 1$.