Section: 17 | Special Functions |

How to Cite this Reference

8 SPECIAL FUNCTIONS

8.1 ORTHOGONAL POLYNOMIALS

Legendre Symbol: $P_{n} (x)$ Interval: [ $- 1, 1$ ]

Differential Equation: $(1 - x^{2}) y^{''} - 2 {x y}^{'} + n (n + 1) y = 0$

Explicit Expression: $P_{n} (x) = \frac{1}{2^{n}} \sum_{m = 0}^{[n / 2]} {(- 1)}^{m} (\begin{matrix} n \\ m \end{matrix}) (\begin{matrix} 2 n - 2 m \\ n \end{matrix}) x^{n - 2 m}$

Recurrence Relation: $(n + 1) P_{n + 1} (x) = (2 n + 1) {x P}_{n} (x) - {n P}_{n - 1} (x)$

Weight: 1

Standardization: $P_{n} (1) = 1$

Norm: $\int_{- 1}^{+ 1} [P_{n} (x)]^{2} d x = \frac{2}{2 n + 1}$

Rodrigues' Formula: $P_{n} (x) = \frac{(- 1)^{n}}{2^{n} n!} \frac{d^{n}}{{d x}^{n}} {(1 - x^{2})^{n}}$

Generating Function: $\begin{matrix} R^{- 1} = \sum_{n = 0}^{\infty} P_{n} (x) z^{n}; - 1 < x < 1, | z | < 1, \\ R = \sqrt{1 - 2 x z + z^{2}} \end{matrix}$

Inequality: $| P_{n} (x) | \leq 1, - 1 \leq x \leq 1$ .
Tschebysheff, First Kind Symbol: $T_{n} (x)$ Interval:[−1, 1]

Differential Equation: $(1 - x^{2}) y - x y^{'} + n^{2} y = 0$

Explicit Expression: $\frac{n}{2} \sum_{m = 0}^{[n / 2]} {(- 1)}^{m} \frac{(n - m - 1)!}{m! (n - 2 m)!} {(2 x)}^{n - 2 m} = cos (n arccos x) = T_{n} (x)$

Recurrence Relation: $T_{n + 1} (x) = 2 {x T}_{n} (x) - T_{n - 1} (x)$

Weight: $(1 - x^{2})^{- 1 / 2}$

Standardization: $T_{n} (1) = 1$

Norm: $\int_{- 1}^{+ 1} (1 - x^{2})^{- 1 / 2} [T_{n} (x)]^{2} d x = {\begin{matrix} π / 2, & n \neq 0 \\ π, & n = 0 \end{matrix}$

Rodrigues' Formula: $\frac{(- 1)^{n} (1 - x^{2})^{1 / 2} \sqrt{π}}{2^{n + 1} Γ (n + \frac{1}{2})} \frac{d^{n}}{{d x}^{n}} {(1 - x^{2})^{n - (1 / 2)}} = T_{n} (x)$

Generating Function: $\frac{1 - x z}{1 - 2 x z - z^{2}} = \sum_{n = 0}^{\infty} T_{n} (x) z^{n}, - 1 < x < 1, | z | < 1$

Inequality: $| T_{n} (x) | \leq 1, - 1 \leq x \leq 1$ .
Tschebysheff, Second Kind Symbol $U_{n} (x)$ Interval: [−1, 1]

Differential Equation: $(1 - x^{2}) y^{″} - 3 x y^{'} + n (n + 2) y = 0$

Explicit Expression: $\begin{matrix} U_{n} (x) = \sum_{m = 0}^{[n / 2]} (- 1)^{m} \frac{(m - n)!}{m! (n - 2 m)!} (2 x)^{n - 2 m} \\ U_{n} (cos θ) = \frac{sin [(n + 1) θ]}{sin θ} \end{matrix}$

Recurrence Relation: $U_{n + 1} (x) = 2 {x U}_{n} (x) - U_{n - 1} (x)$

Weight: $(1 - x^{2})^{1 / 2}$

Standardization: $U_{n} (1) = n + 1$

Norm: $\int_{- 1}^{+ 1} (1 - x^{2})^{1 / 2} [U_{n} (x)]^{2} d x = \frac{π}{2}$

Rodrigues' Formula: $U_{n} (x) = \frac{(- 1)^{n} (n + 1) \sqrt{π}}{(1 - x^{2})^{1 / 2} 2^{n + 1} Γ (n + \frac{3}{2})} \frac{d^{n}}{{d x}^{n}} {(1 - x^{2})^{n + (1 / 2)}}$

Generating Function: $\frac{1}{1 - 2 x z + z^{2}} = \sum_{n = 0}^{\infty} U_{n} (x) z^{n}, - 1 < x < 1, | z | < 1$

Inequality: $| U_{n} (x) | \leq n + 1, - 1 \leq x \leq 1 .$
Jacobi Symbol: $P_{n}^{(α, β)} (x)$ Interval: [−1, 1]

Differential Equation: $(1 - x^{2}) y^{''} + [β - α - (α + β + 2) x] y^{'} + n (n + α + β + 1) y = 0$

Explicit Expression: $P_{n}^{(α, β)} (x) = \frac{1}{2^{n}} \sum_{m = 0}^{n} (\begin{matrix} n + α \\ m \end{matrix}) (\begin{matrix} n + β \\ n - m \end{matrix}) {(x - 1)}^{n - m} {(x + 1)}^{m}$

Recurrence Relation: $\begin{matrix} 2 (n + 1) (n + α + β + 1) (2 n + α + β) P_{n + 1}^{(α, β)} (x) \\ = (2 n + α + β + 1) [(α^{2} - β^{2}) + (2 n + α + β + 2) \\ \times (2 n + α + β) x] P_{n}^{(α, β)} (x) \\ - 2 (n + α) (n + β) (2 n + α + β + 2) P_{n - 1}^{(α, β)} (x) \end{matrix}$

Weight: $(1 - x)^{α} (1 + x)^{β}; α, β > 1$

Standardization: $P_{n}^{(α, β)} (x) = \frac{n + α}{n}$

Norm: $\int_{- 1}^{+ 1} (1 - x)^{α} (1 + x)^{β} [P_{n}^{(α, β)} (x)]^{2} d x = \frac{2^{α + β + 1} Γ (n + α + 1) Γ (n + β + 1)}{(2 n + α + β + 1) n! Γ (n + α + β + 1)}$

Rodrigues' Formula: $P_{n}^{(α, β)} (x) = \frac{(- 1)^{n}}{2^{n} n! (1 - x)^{α} (1 + x)^{β}} \frac{d^{n}}{{d x}^{n}} {(1 - x)^{n + α} (1 + x)^{n + β}}$

Generating Function: $\begin{matrix} R^{- 1} (1 - z + R)^{- α} (1 + z + R)^{- β} = \sum_{n = 0}^{\infty} 2^{- α - β} P_{n}^{(α, β)} (x) z^{n}, \\ R = \sqrt{1 - 2 x z + z^{2}}, | z | < 1 \end{matrix}$

Inequality: $max_{- 1 \leq x \leq 1} | P_{n}^{(α, β)} (x) | = {\begin{array}{l} (\begin{array}{l} n + q \\ n \end{array}) ~ n^{q} if q = max (α, β) \geq - \frac{1}{2} \\ | P_{n}^{(α, β)} (x^{'}) | ~ n^{- 1 / 2} if q < - \frac{1}{2} \\ x^{'} is one of the two maximum points nearest \\ \frac{β - α}{α + β + 1} \end{array}$
Generalized Laguerre Symbol: $L_{n}^{(α)} (x)$ Interval: $[0, \infty]$

Differential Equation: ${x y}^{''} + (α + 1 - x) y^{'} + n y = 0$

Explicit Expression: $L_{n}^{(α)} (x) = \sum_{m = 0}^{n} {(- 1)}^{m} (\begin{matrix} n + α \\ n - m \end{matrix}) \frac{1}{m!} x^{m}$

Recurrence Relation: $(n + 1) L_{n}^{(α)} + 1 (x) = [(2 n + α + 1) - x] L_{n}^{(α)} (x) - (n + α) L_{n}^{(α)} - 1 (x)$

Weight: $x^{α} e^{- x}, α > - 1$

Standardization: $L_{n}^{(α)} (x) = \frac{(- 1)^{n}}{n!} x^{n} + \dots$

Norm: $\int_{0}^{\infty} x^{α} e^{- x} [L_{n}^{(α)} (x)]^{2} d x = \frac{Γ (n + α + 1)}{n!}$

Rodrigues' Formula: $L_{n}^{(α)} (x) = \frac{1}{n! x^{α} e^{- x}} \frac{d^{n}}{{d x}^{n}} {x^{n + α} e^{- x}}$

Generating Function: $(1 - z)^{- α - 1} exp (\frac{x z}{z - 1}) = \sum_{n = 0}^{\infty} L_{n}^{(α)} (x) z^{n}$

Inequality: $\begin{matrix} L_{n}^{(α)} (x) \leq \frac{Γ (n + α + 1)}{n! Γ (α + 1)} e^{x / 2}; \begin{matrix} x \geq 0 \\ α > 0 \end{matrix} \\ | L_{n}^{(a)} (x) | \leq [2 - \frac{Γ (α + n + 1)}{n! Γ (α + 1)}] e^{x / 2}; \begin{matrix} x \geq 0 \\ - 1 < α < 0 \end{matrix} \end{matrix}$
Hermite Symbol: $H_{n} (x)$ Interval: $[- \infty, \infty]$

Differential Equation: $y^{''} - 2 {x y}^{'} + 2 n y = 0$

Explicit Expression: $H_{n} (x) = \sum_{m = 0}^{[n / 2]} \frac{(- 1)^{m} n! (2 x)^{n - 2 m}}{m! (n - 2 m)!}$

Recurrence Relation: $H_{n + 1} (x) = 2 {x H}_{n} (x) - 2 {n H}_{n - 1} (x)$

Weight: $e^{- x^{2}}$

Standardization: $H_{n} (1) = 2^{n} x^{n} + \dots$

Norm: $\int_{- \infty}^{\infty} e^{- x^{2}} {[H_{n} (x)]}^{2} d x = 2^{n} n! \sqrt{π}$

Rodrigues' Formula: $H_{n} (x) = (- 1)^{n} e^{x^{2}} \frac{d^{n}}{{d x}^{n}} (e^{- x^{2}})$

Generating Function: $e^{- x^{2} + 2 z x} = \sum_{n = 0}^{\infty} H_{n} (x) \frac{z^{n}}{n!}$

Inequality: $| H_{n} (x) | e^{x^{2} / 2} k 2^{n / 2} \sqrt{n!} k \approx 1.086435$

8.2 TABLES OF ORTHOGONAL POLYNOMIALS

In the following, ${H_{n}, L_{n}, P_{n}, T_{n}, U_{n}}$ represent the $n^{th}$ order Hermite, Laguerre, Legendre, Tschebysheff (first kind), and Tschebysheff (second kind) polynomials.

$H_{0} = 1$	$x^{10} = (30240 H_{0} + 75600 H_{2} + 25200 H_{4} + 2520 H_{6} + 90 H_{8} + H_{10}) / 1024$
$H_{1} = 2 x$	$x^{9} = (15120 H_{1} + 10080 H_{3} + 1512 H_{5} + 72 H_{7} + H_{9}) / 512$
$H_{2} = 4 x^{2} - 2$	$x^{8} = (1680 H_{0} + 3360 H_{2} + 840 H_{4} + 56 H_{6} + H_{8}) / 256$
$H_{3} = 8 x^{3} - 12 x$	$x^{7} = (840 H_{1} + 420 H_{3} + 42 H_{5} + H_{7}) / 128$
$H_{4} = 16 x^{4} - 48 x^{2} + 12$	$x^{6} = (120 H_{0} + 180 H_{2} + 30 H_{4} + H_{6}) / 64$
$H_{5} = 32 x^{5} - 160 x^{3} + 120 x$	$x^{5} = (60 H_{1} + 20 H_{3} + H_{5}) / 32$
$H_{6} = 64 x^{6} - 480 x^{4} + 720 x^{2} - 120$	$x^{4} = (12 H_{0} + 12 H_{2} + H_{4}) / 16$
$H_{7} = 128 x^{7} - 1344 x^{5} + 3360 x^{3} - 1680 x$	$x^{3} = (6 H_{1} + H_{3}) / 8$
$H_{8} = 256 x^{8} - 3584 x^{6} + 13440 x^{4} - 13440 x^{2} + 1680$	$x^{2} = (2 H_{0} + H_{2}) / 4$
$H_{9} = 512 x^{9} - 9216 x^{7} + 48384 x^{5} - 80640 x^{3} + 30240 x$	$x = (H_{1}) / 2$
$H_{10} = 1024 x^{10} - 23040 x^{8} + 161280 x^{6} - 403200 x^{4} + 302400 x^{2} - 30240$	$1 = H_{0}$
$L_{0} = 1$	$x^{6} = 720 L_{0} - 4320 L_{1} + 10800 L_{2} - 14400 L_{3} + 10800 L_{4} - 4320 L_{5} + 720 L_{6}$
$L_{1} = - x + 1$	$x^{5} = 120 L_{0} - 600 L_{1} + 1200 L_{2} - 1200 L_{3} + 600 L_{4} - 120 L_{5}$
$L_{2} = (x^{2} - 4 x + 2) / 2$	$x^{4} = 24 L_{0} - 96 L_{1} + 144 L_{2} - 96 L_{3} + 24 L_{4}$
$L_{3} = (- x^{3} + 9 x^{2} - 18 x + 6) / 6$	$x^{3} = 6 L_{0} - 18 L_{1} + 18 L_{2} - 6 L_{3}$
$L_{4} = (x^{4} - 16 x^{3} + 72 x^{2} - 96 x + 24) / 24$	$x^{2} = 2 L_{0} - 4 L_{1} + 2 L_{2}$
$L_{5} = (- x^{5} + 25 x^{4} - 200 x^{3} + 600 x^{2} - 600 x + 120) / 120$	$x = L_{0} - L_{1}$
$L_{6} = (x^{6} - 36 x^{5} + 450 x^{4} - 2400 x^{3} + 5400 x^{2} - 4320 x + 720) / 720$	$1 = L_{0}$
$P_{0} = 1$	$x^{10} = (4199 P_{0} + 16150 P_{2} + 15504 P_{4} + 7904 P_{6} + 2176 P_{8} + 256 P_{10}) / 46189$
$P_{1} = x$	$x^{9} = (3315 P_{1} + 4760 P_{3} + 2992 P_{5} + 960 P_{7} + 128 P_{9}) / 12155$
$P_{2} = (3 x^{2} - 1) / 2$	$x^{8} = (715 P_{0} + 2600 P_{2} + 2160 P_{4} + 832 P_{6} + 128 P_{8}) / 6435$
$P_{3} = (5 x^{3} - 3 x) / 2$	$x^{7} = (143 P_{1} + 182 P_{3} + 88 P_{5} + 16 P_{7}) / 429$
$P_{4} = (35 x^{4} - 30 x^{2} + 3) / 8$	$x^{6} = (33 P_{0} + 110 P_{2} + 72 P_{4} + 16 P_{6}) / 231$
$P_{5} = (63 x^{5} - 70 x^{3} + 15 x) / 8$	$x^{5} = (27 P_{1} + 28 P_{3} + 8 P_{5}) / 63$
$P_{6} = (231 x^{6} - 315 x^{4} + 105 x^{2} - 5) / 16$	$x^{4} = (7 P_{0} + 20 P_{2} + 8 P_{4}) / 35$
$P_{7} = (429 x^{7} - 693 x^{5} + 315 x^{3} - 35 x) / 16$	$x^{3} = (3 P_{1} + 2 P_{3}) / 5$
$P_{8} = (6435 x^{8} - 12012 x^{6} + 6930 x^{4} - 1260 x^{2} + 35) / 128$	$x^{2} = (P_{0} + 2 P_{2}) / 3$
$P_{9} = (12155 x^{9} - 25740 x^{7} + 18018 x^{5} - 4620 x^{3} + 315 x) / 128$	$x = P_{1}$
$P_{10} = (46189 x^{10} - 109395 x^{8} + 90090 x^{6} - 30030 x^{4} + 3465 x^{2} - 63) / 256$	$1 = P_{0}$
$T_{0} = 1$	$x^{10} = (126 T_{0} + 210 T_{2} + 120 T_{4} + 45 T_{6} + 10 T_{8} + T_{10}) / 512$
$T_{1} = x$	$x^{9} = (126 T_{1} + 84 T_{3} + 36 T_{5} + 9 T_{7} + T_{9}) / 256$
$T_{2} = 2 x^{2} - 1$	$x^{8} = (35 T_{0} + 56 T_{2} + 28 T_{4} + 8 T_{6} + T_{8}) / 128$
$T_{3} = 4 x^{3} - 3 x$	$x^{7} = (35 T_{1} + 21 T_{3} + 7 T_{5} + T_{7}) / 64$
$T_{4} = 8 x^{4} - 8 x^{2} + 1$	$x^{6} = (10 T_{0} + 15 T_{2} + 6 T_{4} + T_{6}) / 32$
$T_{5} = 16 x^{5} - 20 x^{3} + 5 x$	$x^{5} = (10 T_{1} + 5 T_{3} + T_{5}) / 16$
$T_{6} = 32 x^{6} - 48 x^{4} + 18 x^{2} - 1$	$x^{4} = (3 T_{0} + 4 T_{2} + T_{4}) / 8$
$T_{7} = 64 x^{7} - 112 x^{5} + 56 x^{3} - 7 x$	$x^{3} = (3 T_{1} + T_{3}) / 4$
$T_{8} = 128 x^{8} - 256 x^{6} + 160 x^{4} - 32 x^{2} + 1$	$x^{2} = (T_{0} + T_{2}) / 2$
$T_{9} = 256 x^{9} - 576 x^{7} + 432 x^{5} - 120 x^{3} + 9 x$	$x = T_{1}$
$T_{10} = 512 x^{10} - 1280 x^{8} + 1120 x^{6} - 400 x^{4} + 50 x^{2} - 1$	$1 = T_{0}$
$U_{0} = 1$	$x^{10} = (42 U_{0} + 90 U_{2} + 75 U_{4} + 35 U_{6} + 9 U_{8} + U_{10}) / 1024$
$U_{1} = 2 x$	$x^{9} = (42 U_{1} + 48 U_{3} + 27 U_{5} + 8 U_{7} + U_{9}) / 512$
$U_{2} = 4 x^{2} - 1$	$x^{8} = (14 U_{0} + 28 U_{2} + 20 U_{4} + 7 U_{6} + U_{8}) / 256$
$U_{3} = 8 x^{3} - 4 x$	$x^{7} = (14 U_{1} + 14 U_{3} + 6 U_{5} + U_{7}) / 128$
$U_{4} = 16 x^{4} - 12 x^{2} + 1$	$x^{6} = (5 U_{0} + 9 U_{2} + 5 U_{4} + U_{6}) / 64$
$U_{5} = 32 x^{5} - 32 x^{3} + 6 x$	$x^{5} = (5 U_{1} + 4 U_{3} + U_{5}) / 32$
$U_{6} = 64 x^{6} - 80 x^{4} + 24 x^{2} - 1$	$x^{4} = (2 U_{0} + 3 U_{2} + U_{4}) / 16$
$U_{7} = 128 x^{7} - 192 x^{5} + 80 x^{3} - 8 x$	$x^{3} = (2 U_{1} + U_{3}) / 8$
$U_{8} = 256 x^{8} - 448 x^{6} + 240 x^{4} - 40 x^{2} + 1$	$x^{2} = (U_{0} + U_{2}) / 4$
$U_{9} = 512 x^{9} - 1024 x^{7} + 672 x^{5} - 160 x^{3} + 10 x$	$x = (U_{1}) / 2$
$U_{10} = 1024 x^{10} - 2304 x^{8} + 1792 x^{6} - 560 x^{4} + 60 x^{2} - 1$	$1 = U_{0}$

8.3 BESSEL FUNCTIONS

Bessel's differential equation for a real variable $x$ is $x^{2} \frac{d^{2} y}{{d x}^{2}} + x \frac{d y}{d x} + (x^{2} - n^{2}) y = 0$
When $n$ is not an integer, two independent solutions of the equation are $J_{n} (x)$ and $J_{- n} (x)$ where Jn(x)=∑k=0∞(−1)kk!Γ(n+k+1)(x2)n+2k
If $n$ is an integer then $J_{n} (x) = (- 1)^{n} J_{n} (x)$ , where $J_{n} (x) = \frac{x^{n}}{2^{n} n!} {1 - \frac{x^{2}}{2^{2} \cdot 1! (n + 1)} + \frac{x^{4}}{2^{4} \cdot 2! (n + 1) (n + 2)} + \frac{x^{6}}{2^{6} \cdot 3! (n + 1) (n + 2) (n + 3)} + \dots}$
For $n = 0$ and $n = 1$ , this formula becomes $\begin{matrix} J_{0} (x) & = 1 - \frac{x^{2}}{2^{2} (1!)^{2}} + \frac{x^{4}}{2^{4} (2!)^{2}} - \frac{x^{6}}{2^{6} (3!)^{2}} + \frac{x^{8}}{2^{8} (4!)^{2}} - \dots \\ J_{1} (x) & = \frac{x}{2} - \frac{x^{3}}{2^{3} \cdot 1! 2!} + \frac{x^{5}}{2^{5} \cdot 2! 3!} - \frac{x^{7}}{2^{7} \cdot 3! 4!} + \frac{x^{9}}{2^{9} \cdot 4! 5!} - \dots \end{matrix}$

Table of zeros for

J_{0} (x)

and

J_{1} (x)

. Define

{α_{n}, β_{n}}

J_{0} (α_{n}) = 0

and

J_{1} (β_{n}) = 0

Roots $α_{n}$	$J_{1} (α_{n})$	Roots $β_{n}$	$J_{0} (β_{n})$
2.4048	0.5191	0.0000	1.0000
5.5201	$- 0.3403$	3.8317	$- 0.4028$
8.6537	0.2715	7.0156	0.3001
11.7915	$- 0.2325$	10.1735	$- 0.2497$
14.9309	0.2065	13.3237	0.2184
18.0711	$- 0.1877$	16.4706	$- 0.1965$
21.2116	0.1733	19.6159	0.1801

Recurrence formulas $\begin{matrix} J_{n - 1} (x) + J_{n + 1} (x) = \frac{2 n}{x} J_{n} (x) & {n J}_{n} (x) + {x J}_{n}^{'} (x) = {x J}_{n - 1} (x) \\ J_{n - 1} (x) - J_{n + 1} (x) = 2 J_{n}^{'} (x) & {n J}_{n} (x) - {x J}_{n}^{'} (x) = {x J}_{n + 1} (x) \end{matrix}$
If $J_{n}$ is written for $J_{n} (x)$ and $J_{n}^{(k)}$ is written for $\frac{d^{k}}{{d x}^{k}} {J_{n} (x)}$ , then the following derivative relationships are important $\begin{matrix} J_{0}^{(r)} = - J_{1}^{(r - 1)} \\ J_{0}^{(2)} = - J_{0} + \frac{1}{x} J_{1} = \frac{1}{2} (J_{2} - J_{0}) \\ J_{0}^{(3)} = \frac{1}{x} J_{0} + (1 - \frac{2}{x^{2}}) J_{1} = \frac{1}{4} (- J_{3} + 3 J_{1}) \\ J_{0}^{(4)} = (1 - \frac{3}{x^{2}}) J_{0} - (\frac{2}{x} - \frac{6}{x^{3}}) J_{1} = \frac{1}{8} (J_{4} - 4 J_{2} + 3 J_{0}), etc . \end{matrix}$

Half-order Bessel functions

\begin{matrix} J_{\frac{1}{2}} (x) = \sqrt{\frac{2}{π x}} sin x \\ J_{- \frac{1}{2}} (x) = \sqrt{\frac{2}{π x}} cos x \\ J_{n + \frac{3}{2}} (x) = - x^{n + \frac{1}{2}} \frac{d}{d x} {x^{- (n + \frac{1}{2})} J_{n + \frac{1}{2}} (x)} \\ J_{n - \frac{1}{2}} (x) = x^{- (n + \frac{1}{2})} \frac{d}{d x} {x^{n + \frac{1}{2}} J_{n + \frac{1}{2}} (x)} \end{matrix}

$n$	${(\frac{π x}{2})}^{\frac{1}{2}} J_{n + \frac{1}{2}} (x)$	${(\frac{π x}{2})}^{\frac{1}{2}} J_{- (n + \frac{1}{2})} (x)$
0	$sin x$	$cos x$
1	$\frac{sin x}{x} - cos x$	$- \frac{cos x}{x} - sin x$
2	$(\frac{3}{x^{2}} - 1) sin x - \frac{3}{x} cos x$	$(\frac{3}{x^{2}} - 1) cos x + \frac{3}{x} sin x$
3	$(\frac{15}{x^{3}} - \frac{6}{x}) sin x - (\frac{15}{x^{2}} - 1) cos x$	$- (\frac{15}{x^{3}} - \frac{6}{x}) cos x - (\frac{15}{x^{2}} - 1) sin x$

Additional solutions to Bessel's equation are $Y_{n} (x)$ (also called Weber's function, and sometimes denoted by $N_{n} (x)$ ) and $H_{n}^{(1)} (x)$ and $H_{n}^{(2)} (x)$ (also called Hankel functions) These solutions are defined as follows Yn(x)={Jn(x)cos(nπ)−J−n(x)sin(nπ)nnot an integerHn(1)(x)=Jn(x)+iYn(x)limv→nJv(x)cos(vπ)−J−v(x)sin(vπ)nan integerHn(2)(x)=Jn(x)−iYn(x) The additional properties of these functions may all be derived from the above relations and the known properties of $J_{n} (x)$ .
Complete solutions to Bessel's equation may be written as $c_{1} J_{n} (x) + c_{2} J_{- n} (x)$ when $n$ is not an integer or, for any value of $n$ , $c_{1} J_{n} (x) + c_{2} Y_{n} (x)$ or $c_{1} H_{n}^{(1)} x + c_{2} H_{n}^{(2)} (x)$ .
The modified (or hyperbolic) Bessel's differential equation is $x^{2} \frac{d^{2} y}{{d x}^{2}} + x \frac{d y}{d x} - (x^{2} + n^{2}) y = 0$
When $n$ is not an integer, two independent solutions of the equation are $I_{n} (x)$ and $I_{- n} (x)$ , where $I_{n} (x) = \sum_{k = 0}^{\infty} \frac{1}{k! Γ (n + k + 1)} {(\frac{x}{2})}^{n + 2 k}$
If $n$ is an integer, $I_{n} (x) = I_{- n} (x) = \frac{x^{n}}{2^{n} n!} (1 + \frac{x^{2}}{2^{2} \cdot 1! (n + 1)} + \frac{x^{4}}{2^{4} \cdot 2! (n + 1) (n + 2)} + \frac{x^{6}}{2^{6} \cdot 3! (n + 1) (n + 2) (n + 3)} + \dots)$
For $n = 0$ and $n = 1$ , this formula becomes $\begin{matrix} I_{0} (x) & = 1 + \frac{x^{2}}{2^{2} (1!)^{2}} + \frac{x^{4}}{2^{4} (2!)^{2}} + \frac{x^{6}}{2^{6} (3!)^{2}} + \frac{x^{8}}{2^{8} (4!)^{2}} + \dots \\ I_{1} (x) & = \frac{x}{2} + \frac{x^{3}}{2^{3} \cdot 1! 2!} + \frac{x^{5}}{2^{5} \cdot 2! 3!} + \frac{x^{7}}{2^{7} \cdot 3! 4!} + \frac{x^{9}}{2^{9} \cdot 4! 5!} + \dots \end{matrix}$
Another solution to the modified Bessel's equation is Kn(x)={12πI−n(x)−In(x)sin(nπ)nnot an integerlimv→n12πI−v(x)−Iv(x)sin(vπ)nan integer This function is linearly independent of $I_{n} (x)$ for all values of $n$ . Thus the complete solution to the modified Bessel's equation may be written as $c_{1} I_{n} (x) + c_{2} I_{- n} (x) when n is not an integer$ or $c_{1} I_{n} (x) + c_{2} K_{n} (x) for any value of n$
The following relations hold among the various Bessel functions: $\begin{matrix} I_{n} (z) = i^{- m} J_{m} (i z) \\ Y_{n} (i z) = (i)^{n + 1} I_{n} (z) - \frac{2}{π} i^{- n} K_{n} (z) \end{matrix}$ Most of the properties of the modified Bessel function may be deduced from the known properties of $J_{n} (x)$ by use of these relations and those previously given.
Recurrence formulas $\begin{matrix} I_{n - 1} (x) - I_{n + 1} (x) = \frac{2 n}{x} I_{n} (x) & I_{n - 1} (x) + I_{n + 1} (x) = 2 I_{n}^{'} (x) \\ I_{n - 1} (x) - \frac{n}{x} I_{n} (x) = I_{n}^{'} (x) & I_{n}^{'} (x) = I_{n + 1} (x) + \frac{n}{x} I_{n} (z) \end{matrix}$

8.4 FACTORIAL FUNCTION

For non-negative integers $n$ , the factorial of $n$ , denoted $n!$ , is the product of all positive integers less than or equal to $n$ ; $n! = n \cdot (n - 1) \cdot (n - 2) \dots 2 \cdot 1$ . If $n$ is a negative integer ( $n = - 1, - 2, \dots$ ) then $n! = \pm \infty$ .

Approximations to $n!$ for large $n$ include Stirling's formula $n! \approx \sqrt{2 π e} {(\frac{n}{e})}^{n + \frac{1}{2}},$ and Burnsides's formula $n! \approx \sqrt{2 π} {(\frac{n + \frac{1}{2}}{e})}^{n + \frac{1}{2}} .$

$n$	$n!$	$log_{10} n!$	$n$	$n!$	$log_{10} n!$
$0$	$1$	$0.00000$	$1$	$1$	$0.00000$
$2$	$2$	$0.30103$	$3$	$6$	$0.77815$
$4$	$24$	$1.38021$	$5$	$120$	$2.07918$
$6$	$720$	$2.85733$	$7$	$5040$	$3.70243$
$8$	$40320$	$4.60552$	$9$	$3.6288 \times 10^{5}$	$5.55976$
$10$	$3.6288 \times 10^{6}$	$6.55976$	$11$	$3.9917 \times 10^{7}$	$7.60116$
$12$	$4.7900 \times 10^{8}$	$8.68034$	$13$	$6.2270 \times 10^{9}$	$9.79428$
$14$	$8.7178 \times 10^{10}$	$10.94041$	$15$	$1.3077 \times 10^{12}$	$12.11650$
$16$	$2.0923 \times 10^{13}$	$13.32062$	$17$	$3.5569 \times 10^{14}$	$14.55107$
$18$	$6.4024 \times 10^{15}$	$15.80634$	$19$	$1.2165 \times 10^{17}$	$17.08509$
$20$	$2.4329 \times 10^{18}$	$18.38612$	$25$	$1.5511 \times 10^{25}$	$25.19065$
$30$	$2.6525 \times 10^{32}$	$32.42366$	$40$	$8.1592 \times 10^{47}$	$47.91165$
$50$	$3.0414 \times 10^{64}$	$64.48307$	$60$	$8.3210 \times 10^{81}$	$81.92017$
$70$	$1.1979 \times 10^{100}$	$100.07841$	$80$	$7.1569 \times 10^{118}$	$118.85473$
$90$	$1.4857 \times 10^{138}$	$138.17194$	$100$	$9.3326 \times 10^{157}$	$157.97000$
$110$	$1.5882 \times 10^{178}$	$178.20092$	$120$	$6.6895 \times 10^{198}$	$198.82539$
$130$	$6.4669 \times 10^{219}$	$219.81069$	$150$	$5.7134 \times 10^{262}$	$262.75689$
$500$	$1.2201 \times 10^{1134}$	$1134.0864$	$1000$	$4.0239 \times 10^{2567}$	$2567.6046$

8.5 Gamma Function

Definition: $Γ (n) = \int_{0}^{\infty} t^{n - 1} e^{- t} d t n > 0$

Recursion Formula: $\begin{array}{l} Γ (n + 1) = n Γ (n) \\ Γ (n + 1) = n! if n = 0, 1, 2, \dots where 0! = 1 \\ For n < 0 the gamma function can be defined by using Γ (n) = \frac{Γ (n + 1)}{n} \end{array}$

Special Values: $\begin{matrix} Γ (1 / 2) & = \sqrt{π} \\ Γ (m + \frac{1}{2}) & = \frac{1 \cdot 3 \cdot 5 \dots (2 m - 1)}{2^{m}} \sqrt{π} m = 1, 2, 3, \dots \\ Γ (- m + \frac{1}{2}) & = \frac{(- 1)^{m} 2^{m} \sqrt{π}}{1 \cdot 3 \cdot 5 \dots (2 m - 1)} m = 1, 2, 3, \dots \end{matrix}$

Special Formulas: $\begin{matrix} Γ (x + 1) & = lim_{k \to \infty} \frac{1 \cdot 2 \cdot 3 \dots k}{(x + 1) (x + 2) \dots (x + k)} k^{x} \\ \frac{1}{Γ (x)} & = {x e}^{γ x} Π_{m = 1}^{\infty} {(1 + \frac{x}{m}) e^{- x / m}} (γ is Euler"s constant) \end{matrix}$

Properties: $\begin{matrix} Γ^{'} (1) & = \int_{0}^{\infty} e^{γ x} ln x d x = - γ \\ \frac{Γ^{'} (x)}{Γ (x)} & = - γ + (\frac{1}{1} - \frac{1}{x}) + (\frac{1}{2} - \frac{1}{x + 1}) + \dots + (\frac{1}{n} - \frac{1}{x + n - 1}) + \dots \\ Γ (x + 1) & = \sqrt{2 π x} x^{x} e^{- x} {1 + \frac{1}{12 x} + \frac{1}{288 x^{2}} - \frac{139}{51, 840 x^{3}} + \dots} (Stirling"s asymptotic series) \end{matrix}$

$n$	$Γ (n)$	$n$	$Γ (n)$	$n$	$Γ (n)$	$n$	$Γ (n)$
1.00	1.00000	1.25	.90640	1.50	.88623	1.75	.91906
1.01	.99433	1.26	.90440	1.51	.88659	1.76	.92137
1.02	.98884	1.27	.90250	1.52	.88704	1.77	.92376
1.03	.98355	1.28	.90072	1.53	.88757	1.78	.92623
1.04	.97844	1.29	.89904	1.54	.88818	1.79	.92877
1.05	.97350	1.30	.89747	1.55	.88887	1.80	.93138
1.06	.96874	1.31	.89600	1.56	.88964	1.81	.93408
1.07	.96415	1.32	.89464	1.57	.89049	1.82	.93685
1.08	.95973	1.33	.89338	1.58	.89142	1.83	.93969
1.09	.95546	1.34	.89222	1.59	.89243	1.84	.94261
1.10	.95135	1.35	.89115	1.60	.89352	1.85	.94561
1.11	.94740	1.36	.89018	1.61	.89468	1.86	.94869
1.12	.94359	1.37	.88931	1.62	.89592	1.87	.95184
1.13	.93993	1.38	.88854	1.63	.89724	1.88	.95507
1.14	.93642	1.39	.88785	1.64	.89864	1.89	.95838
1.15	.93304	1.40	.88726	1.65	.90012	1.90	.96177
1.16	.92980	1.41	.88676	1.66	.90167	1.91	.96523
1.17	.92670	1.42	.88636	1.67	.90330	1.92	.96877
1.18	.92373	1.43	.88604	1.68	.90500	1.93	.97240
1.19	.92089	1.44	.88581	1.69	.90678	1.94	.97610
1.20	.91817	1.45	.88566	1.70	.90864	1.95	.97988
1.21	.91558	1.46	.88560	1.71	.91057	1.96	.98374
1.22	.91311	1.47	.88563	1.72	.91258	1.97	.98768
1.23	.91075	1.48	.88575	1.73	.91466	1.98	.99171
1.24	.90852	1.49	.88595	1.74	.91683	1.99	.99581
						2.00	1.00000

8.6 BETA FUNCTION

Definition: $B (m, n) = \int_{0}^{1} t^{m - 1} (1 - t)^{n - 1} d t m > 0, n > 0$

Relationship with Gamma function: $B (m, n) = \frac{Γ (m) Γ (n)}{Γ (m + n)}$

Properties: $\begin{matrix} B (m, n) = B (n, m) \\ B (m, n) = 2 \int_{0}^{π / 2} sin^{2 m - 1} θ cos^{2 n - 1} θ d θ \\ B (m, n) = \int_{0}^{\infty} \frac{t^{m - 1}}{(1 + t)^{m + n}} d t \\ B (m, n) = r^{n} (r + 1)^{m} \int_{0}^{1} \frac{t^{m - 1} (1 - t)^{n - 1}}{(r + t)^{m + n}} d t \end{matrix}$

8.7 ERROR FUNCTION

Definition: $erf (x) = \frac{2}{\sqrt{π}} \int_{0}^{x} e^{- t^{2}} d t$

Series: $erf (x) = \frac{2}{\sqrt{π}} (x - \frac{x^{3}}{3} + \frac{1}{2!} \frac{x^{5}}{5} - \frac{1}{3!} \frac{x^{7}}{7} + \dots)$

Property: $erf (x) = - erf (- x)$

Relationship with Normal Probability Function $f (t)$ : $\int_{0}^{x} f (t) d t = \frac{1}{2} erf (\frac{x}{\sqrt{2}})$ To evaluate $erf (2.3)$ , one proceeds as follows: For $\frac{x}{\sqrt{2}} = 2.3$ , one finds $x = (2.3) (\sqrt{2}) = 3.25$ . In the normal probability function table, one finds the entry 0.4994 opposite the value 3.25. Thus $erf (2.3) = 2 (0.4994) = 0.9988$ . $erfc (z) = 1 - erf (z) = \frac{2}{\sqrt{π}} \int_{z}^{\infty} e^{- t^{2}} d t$ is known as the complementary error function.

Entry Display

This is where the entry will be displayed

8 SPECIAL FUNCTIONS

8.1 ORTHOGONAL POLYNOMIALS

8.2 TABLES OF ORTHOGONAL POLYNOMIALS

8.3 BESSEL FUNCTIONS

8.4 FACTORIAL FUNCTION

8.5 Gamma Function

8.6 BETA FUNCTION

8.7 ERROR FUNCTION

Other ChemNetBase Products

About

Cookie Policy

What are Cookies?

Cookies We Use and Their Purpose

1. Strictly Necessary Cookies

2. Performance Cookies

3. Functionality Cookies

First and Third Party Cookies

Managing Cookies

Changes to Cookie Policy

Questions or Concerns

Taylor and Francis 'First Party' Cookies

'Third Party' Cookies