\form#0:\[ x_{i+1}\leftarrow(ax_{i} + c) \bmod m \] \form#1:$1$ \form#2:\[ x_{i}\leftarrow(x_{i - s} - x_{i - r} - carry_{i-1}) \bmod m \] \form#3:$r$ \form#4:$(m^r - m^s - 1)$ \form#5:\[ p(x|\mu,\sigma) = \frac{1}{\sigma \sqrt{2 \pi}} e^{- \frac{{x - \mu}^ {2}}{2 \sigma ^ {2}} } \] \form#6:\[ p(x|m,s) = \frac{1}{sx\sqrt{2\pi}} \exp{-\frac{(\ln{x} - m)^2}{2s^2}} \] \form#7:\[ p(x|\alpha,\beta) = \frac{1}{\beta\Gamma(\alpha)} (x/\beta)^{\alpha - 1} e^{-x/\beta} \] \form#8:$p(x|n) = \frac{x^{(n/2) - 1}e^{-x/2}}{\Gamma(n/2) 2^{n/2}}$ \form#9:$p(x|a,b) = (\pi b (1 + (\frac{x-a}{b})^2))^{-1}$ \form#10:\[ p(x|m,n) = \frac{\Gamma((m+n)/2)}{\Gamma(m/2)\Gamma(n/2)} (\frac{m}{n})^{m/2} x^{(m/2)-1} (1 + \frac{mx}{n})^{-(m+n)/2} \] \form#11:\[ p(x|n) = \frac{1}{\sqrt(n\pi)} \frac{\Gamma((n+1)/2)}{\Gamma(n/2)} (1 + \frac{x^2}{n}) ^{-(n+1)/2} \] \form#12:$p$ \form#13:$(1 - p)$ \form#14:$p(i|t,p) = \binom{t}{i} p^i (1 - p)^{t - i}$ \form#15:$t$ \form#16:$p(i|p) = p(1 - p)^{i}$ \form#17:$p(i) = \binom{n}{i} p^i (1 - p)^{t - i}$ \form#18:$p(i|\mu) = \frac{\mu^i}{i!} e^{-\mu}$ \form#19:$\mu$ \form#20:$p(x|\lambda) = \lambda e^{-\lambda x}$ \form#21:$\frac{1}{\lambda}$ \form#22:$\frac{\ln 2}{\lambda}$ \form#23:$zero$ \form#24:$[0, \infty]$ \form#25:\[ p(x|\alpha,\beta) = \frac{\alpha}{\beta} (\frac{x}{\beta})^{\alpha-1} \exp{(-(\frac{x}{\beta})^\alpha)} \] \form#26:\[ p(x|a,b) = \frac{1}{b} \exp( \frac{a-x}{b} - \exp(\frac{a-x}{b})) \] \form#27:$x_0$ \form#28:$ m_{lcg} = 2147483563, a_{lcg} = 40014, c_{lcg} = 0, and lcg(0) = value $ \form#29:$ x_{-r} \dots x_{-1} $ \form#30:$ lcg(1) \bmod m \dots lcg(r) \bmod m $ \form#31:$ x_{-1} = 0 $ \form#32:$mean$ \form#33:$\alpha$ \form#34:$\beta$ \form#35:$[0, 1]$ \form#36:$k$ \form#37:$\lambda$ \form#38:$a$ \form#39:$b$ \form#40:\[ \frac{|f - f_{test}|}{|f_{test}|} \] \form#41:$ L_n^m(x) $ \form#42:$ \alpha $ \form#43:$ L_n^\alpha(x) $ \form#44:\[ L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} {}_1F_1(-n; \alpha + 1; x) \] \form#45:$ (\alpha)_n $ \form#46:$ {}_1F_1(a; c; x) $ \form#47:$ \alpha = m $ \form#48:\[ L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) \] \form#49:\[ L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) \] \form#50:$ x >= 0 $ \form#51:$ P_l(x) $ \form#52:\[ P_l^m(x) = (1 - x^2)^{m/2}\frac{d^m}{dx^m}P_l(x) \] \form#53:$ B(a,b) $ \form#54:$B(a,b)$ \form#55:\[ B(a,b) = \int_0^1 t^{a - 1} (1 - t)^{b - 1} dt = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} \] \form#56:$ a > 0 $ \form#57:$ b > 0 $ \form#58:$ E(k) $ \form#59:$ K(k) $ \form#60:\[ K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta} {\sqrt{1 - k^2 sin^2\theta}} \] \form#61:$ F(k,\phi) $ \form#62:$ |k| <= 1 $ \form#63:\[ E(k) = E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta} \] \form#64:$ E(k,\phi) $ \form#65:$ \Pi(k,\nu) $ \form#66:$ \Pi(k,\nu) = \Pi(k,\nu,\pi/2) $ \form#67:\[ \Pi(k,\nu) = \Pi(k,\nu,\pi/2) = \int_0^{\pi/2} \frac{d\theta} {(1 - \nu \sin^2\theta)\sqrt{1 - k^2 \sin^2\theta}} \] \form#68:$ \Pi(k,\nu,\phi) $ \form#69:$ I_{\nu}(x) $ \form#70:$ \nu $ \form#71:\[ I_{\nu}(x) = i^{-\nu}J_\nu(ix) = \sum_{k=0}^{\infty} \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} \] \form#72:$ J_{\nu}(x) $ \form#73:\[ J_{\nu}(x) = \sum_{k=0}^{\infty} \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} \] \form#74:$ K_{\nu}(x) $ \form#75:$ x $ \form#76:\[ K_{\nu}(x) = \frac{\pi}{2} \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi} \] \form#77:$ \nu = n $ \form#78:$ lim_{\nu \to n} $ \form#79:\[ K_{-\nu}(x) = K_{\nu}(x) \] \form#80:$ N_{\nu}(x) $ \form#81:\[ N_{\nu}(x) = \frac{J_{\nu}(x) \cos \nu\pi - J_{-\nu}(x)} {\sin \nu\pi} \] \form#82:$ k $ \form#83:$ \phi $ \form#84:\[ F(k,\phi) = \int_0^{\phi}\frac{d\theta} {\sqrt{1 - k^2 sin^2\theta}} \] \form#85:$ \phi= \pi/2 $ \form#86:\[ E(k,\phi) = \int_0^{\phi} \sqrt{1 - k^2 sin^2\theta} \] \form#87:\[ \Pi(k,\nu,\phi) = \int_0^{\phi} \frac{d\theta} {(1 - \nu \sin^2\theta) \sqrt{1 - k^2 \sin^2\theta}} \] \form#88:$ Ei(x) $ \form#89:\[ Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt \] \form#90:$ H_n(x) $ \form#91:\[ H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} \] \form#92:\[ H_n(-x) = (-1)^n H_n(x) \] \form#93:$ L_n(x) $ \form#94:$ l $ \form#95:$ |x| <= 0 $ \form#96:\[ P_l(x) = \frac{1}{2^l l!}\frac{d^l}{dx^l}(x^2 - 1)^{l} \] \form#97:$ l >= 0 $ \form#98:$ \zeta(s) $ \form#99:$ s $ \form#100:\[ \zeta(s) = \sum_{k=1}^{\infty} k^{-s} \hbox{ for } s > 1 \] \form#101:\[ \zeta(s) = \frac{1}{1-2^{1-s}}\sum_{k=1}^{\infty}(-1)^{k-1}k^{-s} \hbox{ for } 0 <= s <= 1 \] \form#102:\[ \zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s) \] \form#103:$ j_n(x) $ \form#104:\[ j_n(x) = \left(\frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x) \] \form#105:$ \theta $ \form#106:\[ Y_l^m(\theta,\phi) = (-1)^m[\frac{(2l+1)}{4\pi} \frac{(l-m)!}{(l+m)!}] P_l^m(\cos\theta) \exp^{im\phi} \] \form#107:$ n >= 0 $ \form#108:\[ n_n(x) = \left(\frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x) \] \form#109:$ Ai(x) $ \form#110:$ Bi(x) $ \form#111:$ {}_1F_1(a;c;x) $ \form#112:\[ {}_1F_1(a;c;x) = \sum_{n=0}^{\infty} \frac{(a)_n x^n}{(c)_n n!} \] \form#113:$ (x)_k = (x)(x+1)...(x+k-1) $ \form#114:$ (x)_0 = 1 $ \form#115:$ {}_2F_1(a,b;c;x) $ \form#116:\[ {}_2F_1(a;c;x) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n x^n}{(c)_n n!} \] \form#117:$[min, max]$ \form#118:$\sqrt{\mathrm{end} - \mathrm{begin}}$ \form#119:$1/(\mathrm{oversampling} \cdot \mathrm{num\_parts})$