Math F2

John Niclasen

Contents:

1. Special functions (18)

1.1 Legendre functions (18.1)

Legendre's differential equation has the form

\displaylines{(1-x^2)y''-2xy'+l(l+1)y=0}

and has three regular singular points, at x=-1,1,\infty . The parameter l is a given real number, and any solution is called a Legendre function.

For \,\vert\, x\,\vert\, <1 the general solution is

\displaylines{y(x)=c_1y_1(x)+c_2y_2(x)}

1.1.1 Legendre functions for integer l

The first few Legendre polynomials are

\displaylines{P_0(x)=1\cr P_1(x)=x\cr P_2(x)={1\over 2}(3x^2-1)\cr P_3(x)={1\over 2}(5x^3-3x)\cr P_4(x)={1\over 8}(35x^4-30x^2+3)\cr P_5(x)={1\over 8}(63x^5-70x^3+15x)\cr }

The Legendre functions of the second kind is

\displaylines{Q_l(x)=\alpha _ly_2(x)\quad \lor \quad Q_l(x)=\beta _ly_1(x)\cr \alpha _l={(-1)^{l/2}2^l\lbrack (l/2)!\rbrack ^2\over l!}\quad for l even\cr \beta _l={(-1)^{(l+1)/2}2^{l-1}\lbrace \lbrack (l-1)/2\rbrack !\rbrace ^2\over l!}\quad for l odd\cr }

The general solution for integer l is

\displaylines{y(x)=c_1P_l(x)+c_2Q_l(x)}

where P_l(x) is a polynomial of order l, and so converges for all x, and Q_l(x) is an infinite series that converges only for \,\vert\, x\,\vert\, <1.

1.1.2 Rodrigues' formula

\displaylines{P_l(x)={1\over 2^ll!} {d^l\over dx^l}(x^2-l)^l}

1.1.3 Mutual orthogonality

\displaylines{\int _{-1}^1P_l(x)P_k(x) dx=0\quad ,\quad if l\not =k}

Any reasonable function f(x) can be expressed in the interval \,\vert\, x\,\vert\, <1 as an infinite sum of Legendre polynomials

\displaylines{f(x)=\sum _{l=0}^{\infty }a_lP_l(x)\cr a_l={2l+1\over 2}\int _{-1}^1f(x)P_l(x) dx\cr }

1.1.4 Generating function

\displaylines{G(x,h)=\sum _{n=0}^{\infty }f_n(x)h^n}

The function P_n(x) defined by the equation

\displaylines{G(x,h)=(1-2xh+h^2)^{-1/2}=\sum _{n=0}^{\infty }P_n(x)h^n}

satisfy Legendre's equation.

1.1.5 Recurrence relations

\displaylines{P'_{n+1}+P'_{n-1}=P_n+2xP'_n\cr P'_{n+1}=(n+1)P_n+xP'_n\cr P'_{n-1}=-nP_n+xP'_n\cr (1-x^2)P'_n=n(P_{n-1}-xP_n)\cr (2n+1)P_n=P'_{n+1}-P'_{n-1}\cr (n+1)P_{n+1}=(2n+1)xP_n-nP_{n-1}\cr }

1.2 Associated Legendre functions (18.2)

The associated Legendre equation has the form

\displaylines{(1-x^2)y''-2xy'+\left [l(l+1)-{m^2\over 1-x^2}\right ]y=0}

If u(x) is a solution of Legendre's equation, then

\displaylines{y(x)=(1-x^2)^{\,\vert\, m\,\vert\, /2}{d^{\,\vert\, m\,\vert\, }u\over dx^{\,\vert\, m\,\vert\, }}}

is a solution of the associated equation.

1.2.1 General solution

\displaylines{y(x)=c_1y_1(x)+c_2y_2(x)}

1.2.2 Associated Legendre functions for integer l

\displaylines{y(x)=c_1P_l^m(x)+c_2Q_l^m(x)}

For m\ge 0

\displaylines{P_l^m(x)=(1-x^2)^{m/2}{d^mP_l\over dx^m}\cr Q_l^m(x)=(1-x^2)^{m/2}{d^mQ_l\over dx^m}\cr }

For -l\le m\le l

\displaylines{P_l^{-m}(x)=(-1)^m{(l-m)!\over (l+m)!}P_l^m(x)}

The first few associated Legendre functions of the first kind

\displaylines{P_1^1(x)=(1-x^2)^{1/2}\cr P_2^1(x)=3x(1-x^2)^{1/2}\cr P_3^2(x)=3(1-x^2)\cr P_3^1(x)={3\over 2}(5x^2-1)(1-x^2)^{1/2}\cr P_3^2(x)=15x(1-x^2)\cr P_3^3(x)=15(1-x^2)^{3/2}\cr }

The associated Legendre functions of the second kind Q_l^m(x), like Q_l(x), are singular at x=\pm 1.

1.2.3 Properties of associated Legendre functions

See p. 589-590

1.2.4 Mutual orthogonality

\displaylines{\int _{-1}^1P_l^m(x)P_k^m(x) dx=0\quad if l\not =k}

When l=k

\displaylines{I_{lm}\equiv \int _{-1}^1P_l^m(x)P_l^m(x) dx={2\over 2l+1} {(l+m)!\over (l-m)!}}

1.2.5 Generating function

\displaylines{G(x,h)={(2m)!(1-x^2)^{m/2}\over 2^mm!(1-2hx+h^2)^{m+1/2}}=\sum _{n=0}^{\infty }P_{n+m}^m(x)h^n}

1.2.6 Recurrence relations

\displaylines{P_n^{m+1}={2mx\over (1-x^2)^{1/2}}P_n^m+\lbrack m(m-1)-n(n+1)\rbrack P_n^{m-1}\cr (2n+1)(1-x^2)^{1/2}P_n^m=(n+m)P_{n-1}^m+(n-m+1)P_{n+1}^m\cr (2n+1)(1-x^2)^{1/2}P_n^m=P_{n+1}^{m+1}-P_{n-1}^{m+1}\cr 2(1-x^2)^{1/2}(P_n^m)'=P_n^{m+1}-(n+m)(n-m+1)P_n^{m-1}\cr }

1.3 Spherical harmonics (18.3)

\displaylines{\nabla ^2u=0\cr \Theta (\theta )\Phi (\phi )=P_l^m(\cos  \theta )(C \cos  m\phi +D \sin  m\phi )\cr }

1.3.1 Spherical harmonics

\displaylines{Y_l^m(\theta ,\phi )=(-1)^m\left [{2l+1\over 4\pi } {(l-m)!\over (l+m)!}\right ]^{1/2}P_l^m(\cos  \theta ) \exp (im\phi )\cr Y_l^{-m}(\theta ,\phi )=(-1)^m\lbrack Y_l^m(\theta ,\phi )\rbrack ^{\ast }\cr }

The first few sperical harmonics Y_l^m(\theta ,\phi )\equiv Y_l^m are as follows

\displaylines{Y_0^0=\sqrt {{1\over 4\pi }}\cr Y_1^0=\sqrt {{3\over 4\pi }}\cos  \theta \cr Y_1^{\pm 1}=\mp \sqrt {{3\over 8\pi }}\sin  \theta  \exp (\pm i\phi )\cr Y_2^0=\sqrt {{5\over 16\pi }}(3 \cos ^2 \theta -1)\cr Y_2^{\pm 1}=\mp \sqrt {{15\over 8\pi }}\sin  \theta  \cos  \theta  \exp (\pm i\phi )\cr Y_2^{\pm 2}=\sqrt {{15\over 32\pi }}\sin ^2 \theta  \exp (\pm 2i\phi )\cr }

1.4 Bessel functions (18.5)

\displaylines{x^2y''+xy'+(x^2-v^2)y=0}

1.4.1 Bessel functions for non-integer v

General solution

\displaylines{y(x)=c_1J_v(x)+c_2J_{-v}(x)\cr J_v(x)=\sum _{n=0}^{\infty }{(-1)^n\over n!\Gamma (v+n+1)}\left ({2\over 2}\right )^{v+2n}\cr }

where \Gamma (x) is the gamma function.

1.4.2 Bessel functions for integer v

General solution

\displaylines{Y_v(x)={J_v(x) \cos  v\pi -J_{-v}(x)\over \sin  v\pi }\cr J_{-v}(x)=(-1)^vJ_v(x)\cr }

1.4.3 Properties of Bessel functions

See p. 608

1.4.4 Mutual orthogonality

\displaylines{\int _a^bxJ_v(\alpha x)J_v(\beta x) dx=0\quad ,\quad \alpha \not =\beta }

1.4.5 Recurrence relations

\displaylines{{d\over dx}\lbrack x^vJ_v(x)\rbrack =x^vJ_{v-1}(x)\cr {d\over dx}\lbrack x^{-v}J_v(x)\rbrack =-x^{-v}J_{v+1}(x)\cr xJ_v'(x)+vJ_v(x)=xJ_{v-1}(x)\cr xJ_v'(x)-vJ_v(x)=-xJ_{v+1}(x)\cr J_{v-1}(x)-J_{v+1}(x)=2J'_v(x)\cr J_{v-1}(x)+J_{v+1}(x)={2v\over x}J_v(x)\cr }

The first two in integral form

\displaylines{\int x^vJ_{v-1}(x) dx=x^vJ_v(x)\cr \int x^{-v}J_{v+1}(x) dx=-x^{-v}J_v(x)\cr }

1.4.6 Generating function

\displaylines{G(x,h)=\exp \left [{x\over 2}\left (h-{1\over h}\right )\right ]=\sum _{n=-\infty }^{\infty }J_n(x)h^n}

1.4.7 Integral representations

\displaylines{J_n(x)={1\over \pi }\int _0^{\pi }\cos (n\theta -x \sin  \theta ) d\theta }

Special case for n=0

\displaylines{J_0(x)={1\over \pi }\int _0^{\pi }\cos (x \sin  \theta ) d\theta ={1\over 2\pi }\int _0^{2\pi }\cos (x \sin  \theta ) d\theta \cr {1\over 2\pi }\int _0^{2\pi }\sin (x \sin  \theta ) d\theta =0\cr \Rightarrow \quad J_0(x)={1\over 2\pi }\int _0^{2\pi }\exp (ix \sin  \theta ) d\theta ={1\over 2\pi }\int _0^{2\pi }\exp (ix \cos  \theta ) d\theta \cr }

1.5 Spherical Bessel functions (18.6)

Helmholtz' equation

\displaylines{(\nabla ^2+k^2)u=0}

in spherical polar coordinates. The radial part R(r) of the solution must satisfy

\displaylines{r^2R''+2rR'+\lbrack k^2r^2-l(l+1)\rbrack R=0}

General solution

\displaylines{R(r)=r^{-1/2}\lbrack c_1J_{l+1/2}(kr)+c_2Y_{l+1/2}(kr)\rbrack }

The functions x^{-1/2}J_{l+1/2}(x) and x^{-1/2}Y_{l+1/2}(x), when suitably normalised, are called spherical Bessel functions of the first and second kind, erspectively, and are denoted as follows:

\displaylines{j_l(x)=\sqrt {{\pi \over 2x}}J_{l+1/2}(x)\cr n_l(x)=\sqrt {{\pi \over 2x}}Y_{l+1/2}(x)\cr }

For l=0

\displaylines{j_0(x)={\sin  x\over x}\cr n_0(x)=-{\cos  x\over x}\cr }

1.6 The gamma function and related functions (18.12)

1.6.1 The gamma function

\displaylines{\Gamma (n)=\int _0^{\infty }x^{n-1}e^{-x} dx\cr \Gamma (n+1)=n\int _0^{\infty }x^{n-1}e^{-x} dx\cr \Gamma (n+1)=n\Gamma (n)\cr \Gamma (1)=1\cr \Gamma (n+1)=n!\quad n\in \mathbb{N} _+\cr \Gamma \left ({1\over 2}\right )=\sqrt {\pi }\cr \Gamma (n)\Gamma (1-n)={\pi \over \sin  n\pi }\cr }

1.6.2 Stirling's approximation

\displaylines{n!\approx \sqrt {2\pi n}n^ne^{-n}}

1.6.3 The beta function

\displaylines{B(m,n)=\int _0^1x^{m-1}(1-x)^{n-1} dx\cr B(m,n)={\Gamma (m)\Gamma (n)\over \Gamma (m+n)}\cr }

1.6.4 The incomplete gamma function

\displaylines{\Gamma (n)=\int _0^xu^{n-1}e^{-u} du+\int _x^{\infty }u^{n-1}e^{-u} du\equiv \gamma (n,x)+\Gamma (n,x)}

1.6.5 The error function

\displaylines{erf(x)={\gamma (1/2,x^2)\over \sqrt {\pi }}\cr erf(x)={2\over \sqrt {\pi }}\int _0^xe^{-u^2} du=1-{2\over \sqrt {\pi }}\int _x^{\infty }e^{-u^2} du\cr erf(0)=0\cr erf(\infty )=1\cr erf(-x)=-erf(x)\cr }