Special Functions

Cheatsheet Content

### Gamma Function - **Definition:** $\Gamma(z) = \int_0^\infty t^{z-1}e^{-t} dt$, for $\text{Re}(z) > 0$. - **Recurrence Relation:** $\Gamma(z+1) = z\Gamma(z)$. - **Special Values:** - $\Gamma(n+1) = n!$ for non-negative integer $n$. - $\Gamma(1) = 1$. - $\Gamma(1/2) = \sqrt{\pi}$. - **Reflection Formula:** $\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin(\pi z)}$. - **Weierstrass Product Formula:** $\frac{1}{\Gamma(z)} = z e^{\gamma z} \prod_{n=1}^\infty \left(1 + \frac{z}{n}\right)e^{-z/n}$, where $\gamma$ is the Euler-Mascheroni constant. - **Beta Function:** $B(x, y) = \int_0^1 t^{x-1}(1-t)^{y-1} dt = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$. ### Bessel Functions - **Bessel's Differential Equation:** $x^2 y'' + xy' + (x^2 - \nu^2)y = 0$. - **Bessel Function of the First Kind ($J_\nu(x)$):** - **Series Representation:** $J_\nu(x) = \sum_{k=0}^\infty \frac{(-1)^k}{k! \Gamma(\nu+k+1)} \left(\frac{x}{2}\right)^{2k+\nu}$. - **Integral Representation:** $J_n(x) = \frac{1}{\pi} \int_0^\pi \cos(x\sin\theta - n\theta) d\theta$ for integer $n$. - **Properties:** - $J_{-n}(x) = (-1)^n J_n(x)$ for integer $n$. - $J_0(0) = 1$, $J_n(0) = 0$ for $n \ne 0$. - $\frac{d}{dx}[x^\nu J_\nu(x)] = x^\nu J_{\nu-1}(x)$. - $\frac{d}{dx}[x^{-\nu} J_\nu(x)] = -x^{-\nu} J_{\nu+1}(x)$. - **Bessel Function of the Second Kind ($Y_\nu(x)$ or $N_\nu(x)$):** - **Definition:** $Y_\nu(x) = \frac{J_\nu(x)\cos(\nu\pi) - J_{-\nu}(x)}{\sin(\nu\pi)}$. For integer $\nu$, it's defined by a limit. - **General Solution:** $y(x) = c_1 J_\nu(x) + c_2 Y_\nu(x)$. - **Modified Bessel Functions:** - **Equation:** $x^2 y'' + xy' - (x^2 + \nu^2)y = 0$. - **First Kind ($I_\nu(x)$):** $I_\nu(x) = i^{-\nu} J_\nu(ix) = \sum_{k=0}^\infty \frac{1}{k! \Gamma(\nu+k+1)} \left(\frac{x}{2}\right)^{2k+\nu}$. - **Second Kind ($K_\nu(x)$):** $K_\nu(x) = \frac{\pi}{2} \frac{I_{-\nu}(x) - I_\nu(x)}{\sin(\nu\pi)}$. ### Legendre Polynomials - **Legendre's Differential Equation:** $(1-x^2)y'' - 2xy' + n(n+1)y = 0$. - **Rodrigues' Formula:** $P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n}(x^2-1)^n$. - **Generating Function:** $\frac{1}{\sqrt{1-2xt+t^2}} = \sum_{n=0}^\infty P_n(x)t^n$. - **Orthogonality:** $\int_{-1}^1 P_m(x)P_n(x) dx = \frac{2}{2n+1}\delta_{mn}$. - **Recurrence Relations:** - $(n+1)P_{n+1}(x) = (2n+1)xP_n(x) - nP_{n-1}(x)$. - $nP_n(x) = xP_n'(x) - P_{n-1}'(x)$. - $P_{n+1}'(x) - P_{n-1}'(x) = (2n+1)P_n(x)$. - **First Few Polynomials:** - $P_0(x) = 1$ - $P_1(x) = x$ - $P_2(x) = \frac{1}{2}(3x^2-1)$ - $P_3(x) = \frac{1}{2}(5x^3-3x)$ ### Hermite Polynomials - **Hermite's Differential Equation:** $y'' - 2xy' + 2ny = 0$. - **Rodrigues' Formula:** $H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n}e^{-x^2}$. - **Generating Function:** $e^{2xt-t^2} = \sum_{n=0}^\infty \frac{H_n(x)}{n!}t^n$. - **Orthogonality:** $\int_{-\infty}^\infty e^{-x^2} H_m(x)H_n(x) dx = \sqrt{\pi} 2^n n! \delta_{mn}$. - **Recurrence Relations:** - $H_{n+1}(x) = 2xH_n(x) - 2nH_{n-1}(x)$. - $H_n'(x) = 2nH_{n-1}(x)$. - **First Few Polynomials:** - $H_0(x) = 1$ - $H_1(x) = 2x$ - $H_2(x) = 4x^2-2$ - $H_3(x) = 8x^3-12x$ ### Laguerre Polynomials - **Laguerre's Differential Equation:** $xy'' + (1-x)y' + ny = 0$. - **Rodrigues' Formula:** $L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n}(x^n e^{-x})$. - **Generating Function:** $\frac{e^{-xt/(1-t)}}{1-t} = \sum_{n=0}^\infty L_n(x)t^n$. - **Orthogonality:** $\int_0^\infty e^{-x} L_m(x)L_n(x) dx = \delta_{mn}$. - **Recurrence Relations:** - $(n+1)L_{n+1}(x) = (2n+1-x)L_n(x) - nL_{n-1}(x)$. - $L_n'(x) = \sum_{k=0}^{n-1} (-1)^{n-k-1} L_k(x)$. - **First Few Polynomials:** - $L_0(x) = 1$ - $L_1(x) = 1-x$ - $L_2(x) = \frac{1}{2}(x^2-4x+2)$ ### Hypergeometric Functions - **Hypergeometric Differential Equation:** $z(1-z)w'' + [c-(a+b+1)z]w' - abw = 0$. - **Series Solution (Gauss Hypergeometric Function):** - $F(a,b;c;z) = {}_2F_1(a,b;c;z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n} \frac{z^n}{n!}$ - **Pochhammer Symbol:** $(q)_n = q(q+1)...(q+n-1) = \frac{\Gamma(q+n)}{\Gamma(q)}$. - **Special Cases:** Many common functions are hypergeometric: - $(1-z)^{-a} = F(a,1;1;z)$ - $\ln(1+z) = z F(1,1;2;-z)$ - $\arcsin(z) = z F(1/2,1/2;3/2;z^2)$ - Legendre Polynomials: $P_n(x) = F(-n, n+1; 1; (1-x)/2)$ - **Confluent Hypergeometric Function ($M(a,c,z)$ or ${}_1F_1(a;c;z)$):** - **Equation:** $zw'' + (c-z)w' - aw = 0$. - **Series:** $M(a,c,z) = \sum_{n=0}^\infty \frac{(a)_n}{(c)_n} \frac{z^n}{n!}$. - Related to Bessel, Hermite, Laguerre functions. ### Summary of Orthogonal Polynomials | Polynomial | Eq. Type | Weight Function $w(x)$ | Interval | Norm $\int w(x) P_n^2(x) dx$ | |---|---|---|---|---| | **Legendre $P_n(x)$** | $(1-x^2)y''-2xy'+n(n+1)y=0$ | $1$ | $[-1,1]$ | $\frac{2}{2n+1}$ | | **Hermite $H_n(x)$** | $y''-2xy'+2ny=0$ | $e^{-x^2}$ | $(-\infty,\infty)$ | $\sqrt{\pi} 2^n n!$ | | **Laguerre $L_n(x)$** | $xy''+(1-x)y'+ny=0$ | $e^{-x}$ | $[0,\infty)$ | $1$ | | **Associated Laguerre $L_n^{(\alpha)}(x)$** | $xy''+(\alpha+1-x)y'+ny=0$ | $x^\alpha e^{-x}$ | $[0,\infty)$ | $\frac{\Gamma(n+\alpha+1)}{n!}$ | | **Chebyshev $T_n(x)$** | $(1-x^2)y''-xy'+n^2y=0$ | $(1-x^2)^{-1/2}$ | $[-1,1]$ | $\frac{\pi}{2}$ (for $n>0$), $\pi$ (for $n=0$) | ### Dirac Delta Function - **Definition:** $\delta(x) = 0$ for $x \ne 0$, and $\int_{-\infty}^\infty \delta(x) dx = 1$. - **Sifting Property:** $\int_{-\infty}^\infty f(x)\delta(x-a) dx = f(a)$. - **Derivative:** $\int_{-\infty}^\infty f(x)\delta'(x-a) dx = -f'(a)$. - **Fourier Transform:** $\mathcal{F}\{\delta(x)\} = 1$. ### Fourier Series and Transforms - **Fourier Series (Period $2L$):** - $f(x) = a_0 + \sum_{n=1}^\infty (a_n \cos(\frac{n\pi x}{L}) + b_n \sin(\frac{n\pi x}{L}))$ - $a_0 = \frac{1}{2L}\int_{-L}^L f(x) dx$ - $a_n = \frac{1}{L}\int_{-L}^L f(x) \cos(\frac{n\pi x}{L}) dx$ - $b_n = \frac{1}{L}\int_{-L}^L f(x) \sin(\frac{n\pi x}{L}) dx$ - **Complex Fourier Series:** $f(x) = \sum_{n=-\infty}^\infty c_n e^{i n\pi x / L}$, where $c_n = \frac{1}{2L}\int_{-L}^L f(x) e^{-i n\pi x / L} dx$. - **Fourier Transform:** - $\mathcal{F}\{f(x)\} = F(\omega) = \int_{-\infty}^\infty f(x) e^{-i\omega x} dx$. - $\mathcal{F}^{-1}\{F(\omega)\} = f(x) = \frac{1}{2\pi}\int_{-\infty}^\infty F(\omega) e^{i\omega x} d\omega$. - **Properties:** - Linearity: $\mathcal{F}\{af(x) + bg(x)\} = aF(\omega) + bG(\omega)$. - Time Shift: $\mathcal{F}\{f(x-a)\} = e^{-i\omega a} F(\omega)$. - Frequency Shift: $\mathcal{F}\{e^{iax}f(x)\} = F(\omega-a)$. - Derivative: $\mathcal{F}\{f'(x)\} = i\omega F(\omega)$. - Convolution: $\mathcal{F}\{f(x)*g(x)\} = F(\omega)G(\omega)$.