0

1

2

3

$\Beta(x, y) = \Beta(y, x)$

$\Beta(x, y) = \Beta(1-x, 1-y)$

$\Beta(x, y) = \Gamma(x) \Gamma(y)$

$\Beta(x, y) = \frac{\Gamma(x) \Gamma(y)}{\Gamma(x+y)}$

0

1

2

3

They are orthogonal on the interval ([-1, 1])

They satisfy the differential equation ((1-x^2)y'' - 2xy' + n(n+1)y = 0)

They are complete on the interval ([-1, 1])

All of the above

$\nu$

$\nu+1$

$\nu-1$

$\nu+2$

It is related to the Bessel function (J_\nu(x)) by (I_\nu(x) = i^{-\nu} J_\nu(ix))

It satisfies the differential equation (x^2y'' + xy' - (x^2 + \nu^2)y = 0)

It has the asymptotic expansion (I_\nu(x) \sim \frac{1}{\sqrt{2\pi x}} e^x) as (x \to \infty)

All of the above

$\sqrt{\pi}$

$\frac{1}{\sqrt{\pi}}$

$\frac{\pi}{2}$

$\frac{2}{\pi}$

$\Beta(x, y) = \Beta(1-x, y)$

$\Beta(x, y) = \Beta(x, 1-y)$

$\Beta(x, y) = \Beta(y, 1-x)$

All of the above

0

1

2

3

They are orthogonal on the interval ([0, 1])

They satisfy the differential equation ((1-x^2)y'' - 2xy' + n(n+1)y = 0)

They are complete on the interval ([0, 1])

All of the above

$\nu$

$\nu+1$

$\nu-1$

$\nu+2$

It is related to the Bessel function (J_\nu(x)) by (K_\nu(x) = \frac{\pi}{2} i^{\nu+1} H_\nu^{(1)}(ix))

It satisfies the differential equation (x^2y'' + xy' - (x^2 + \nu^2)y = 0)

It has the asymptotic expansion (K_\nu(x) \sim \sqrt{\frac{\pi}{2x}} e^{-x}) as (x \to \infty)

All of the above

$\frac{\sqrt{\pi}}{2}$

$\frac{2}{\sqrt{\pi}}$

$\frac{\pi}{2}$

$\frac{3\sqrt{\pi}}{2}$

$\Beta(x, y) = \Beta(x+1, y-1)$

$\Beta(x, y) = \Beta(x-1, y+1)$

$\Beta(x, y) = \Beta(y+1, x-1)$

All of the above

0

1

2

3