$J_0(x)$ $x=5$ $e$ $\log_2 (e)$ $\log_10 (e)$ $\sqrt 2$ $\sqrt@{1/2@}$ $\sqrt 3$ $\pi$ $\pi/2$ $\pi/4$ $\sqrt\pi$ $2/\sqrt\pi$ $1/\pi$ $2/\pi$ $\ln(10)$ $\ln(2)$ $\ln(\pi)$ $\gamma$ $+\infty$ $-\infty$ $+1$ $-1$ $\log(1+x)$ $\exp(x)-1$ $\sqrt@{x^2 + y^2@}$ $\arccosh(x)$ $\arcsinh(x)$ $\arctanh(x)$ $x * 2^e$ $x$ $f$ $x = f * 2^e$ $0.5 <= f < 1$ $x^n$ $x^8$ $((x^2)^2)^2$ $x^2$ $x^3$ $y$ $2 \delta$ $\delta = 2^k \epsilon$ $k$ $x < y$ $x > y$ $z = x + i y$ $z = r \exp(i \theta) = r (\cos(\theta) + i \sin(\theta))$ $3 + 4i$ $\arg(z)$ $-\pi < \arg(z) <= \pi$ $\vert z\vert$ $\vert z\vert^2$ $\log\vert z\vert$ $z=a+b$ $z=a-b$ $z=ab$ $z=a/b$ $z=a+x$ $z=a-x$ $z=ax$ $z=a/x$ $i$ $z=a+iy$ $z=a-iy$ $z=a*(iy)$ $z=a/(iy)$ $z^* = x - i y$ $1/z = (x - i y)/(x^2 + y^2)$ $-z = (-x) + i(-y)$ $\sqrt z$ $z^a$ $\exp(\log(z)*a)$ $z^x$ $\exp(z)$ $\log(z)$ $\log_10 (z)$ $\log_b(z)$ $\log(z)/\log(b)$ $\sin(z) = (\exp(iz) - \exp(-iz))/(2i)$ $\cos(z) = (\exp(iz) + \exp(-iz))/2$ $\tan(z) = \sin(z)/\cos(z)$ $\sec(z) = 1/\cos(z)$ $\csc(z) = 1/\sin(z)$ $\cot(z) = 1/\tan(z)$ $\arcsin(z)$ $1$ $z$ $[-\pi/2,\pi/2]$ $-\pi/2$ $\arccos(z)$ $[0,\pi]$ $\arctan(z)$ $-i$ $\arcsec(z) = \arccos(1/z)$ $\arccsc(z) = \arcsin(1/z)$ $\arccot(z) = \arctan(1/z)$ $\sinh(z) = (\exp(z) - \exp(-z))/2$ $\cosh(z) = (\exp(z) + \exp(-z))/2$ $\tanh(z) = \sinh(z)/\cosh(z)$ $\sech(z) = 1/\cosh(z)$ $\csch(z) = 1/\sinh(z)$ $\coth(z) = 1/\tanh(z)$ $\arcsinh(z)$ $\arccosh(z)$ $\arccosh(z)=\log(z-\sqrt@{z^2-1@})$ $\arctanh(z)$ $\arcsech(z) = \arccosh(1/z)$ $\arccsch(z) = \arcsin(1/z)$ $\arccoth(z) = \arctanh(1/z)$ $c[0] + c[1] x + c[2] x^2 + \dots + c[len-1] x^@{len-1@}$ $a=0$ $(x-1)^2=0$ $b^2 - 4 a c$ $(x-1)^3=0$ $P(x) = a_0 + a_1 x + a_2 x^2 + ... + a_@{n-1@} x^@{n-1@}$ $n-1$ $2(n-1)$ $P(x) = x^5 - 1$ $z_n = \exp(2 \pi n i/5)$ $2 * 10^-16$ $10^-7$ $5 * 10^-4$ $Ai(x)$ $Bi(x)$ $S_A(x) Ai(x)$ $x>0$ $S_A(x)$ $\exp(+(2/3) x^(3/2))$ $x<0$ $S_B(x) Bi(x)$ $S_B(x)$ $exp(-(2/3) x^(3/2))$ $Ai'(x)$ $Bi'(x)$ $S_A(x) Ai'(x)$ $S_B(x) Bi'(x)$ $J_n(x)$ $Y_n(x)$ $I_n(x)$ $K_n(x)$ $j_l(x)$ $y_l(x)$ $i_l(x)$ $k_l(x)$ $J_1(x)$ $n$ $Y_0(x)$ $Y_1(x)$ $I_0(x)$ $I_1(x)$ $\exp(-\vert x\vert) I_0(x)$ $\exp(-\vert x\vert) I_1(x)$ $\exp(-\vert x\vert) I_n(x)$ $K_0(x)$ $x>0$ $K_1(x)$ $\exp(x) K_0(x)$ $\exp(x) K_1(x)$ $\exp(x) K_n(x)$ $j_0(x) = \sin(x)/x$ $j_1(x) = (\sin(x)/x - \cos(x))/x$ $j_2(x) = ((3/x^2 - 1)\sin(x) - 3\cos(x)/x)/x$ $l >= 0$ $x >= 0$ $l$ $lmax >= 0$ $y_0(x) = -\cos(x)/x$ $y_1(x) = -(\cos(x)/x + \sin(x))/x$ $y_2(x) = (-3/x^3 + 1/x)\cos(x) - (3/x^2)\sin(x)$ $i_l(x) = \sqrt@{\pi/(2x)@} I_@{l+1/2@}(x)$ $\exp(-\vert x\vert) i_0(x)$ $\exp(-\vert x\vert) i_1(x)$ $\exp(-\vert x\vert) i_2(x)$ $\exp(-\vert x\vert) i_l(x)$ $\exp(-\vert x\vert) i_l(x)$ $k_l(x) = \sqrt@{\pi/(2x)@} K_@{l+1/2@}(x)$ $\exp(x) k_0(x)$ $\exp(x) k_1(x)$ $\exp(x) k_2(x)$ $\exp(x) k_l(x)$ $\nu$ $J_\nu(x)$ $J_\nu(x_i)$ $Y_\nu(x)$ $I_\nu(x)$ $\nu>0$ $\exp(-\vert x\vert)I_\nu(x)$ $K_\nu(x)$ $\ln(K_\nu(x))$ $\exp(+\vert x\vert) K_\nu(x)$ $Cl_2(\theta) = \Im Li_2(\exp(i\theta))$ $Cl_2(x)$ $R_1 := 2Z \sqrt@{Z@} \exp(-Z r)$ $L^a_b(x)$ $\psi$ $\psi(n,l,r) = R_n Y_@{lm@}$ $F_L(\eta,x)$ $G_L(\eta,x)$ $G_@{L-k@}(\eta,x)$ $F'_L(\eta,x)$ $G'_@{L-k@}(\eta,x)$ $L, L-k > -1/2$ $L$ $L = Lmin \dots Lmin + kmax$ $G'_L(\eta,x)$ $F_L(\eta, x)/x$ $\eta \to 0$ $C_L(\eta)$ $L > -1$ $Lmin > -1$ $ja$ $ma$ $\exp(-x^2) \int_0^x dt \exp(t^2)$ $D_n(x)$ $D_1(x) = (1/x) \int_0^x dt (t/(e^t - 1))$ $D_2(x) = (2/x^2) \int_0^x dt (t^2/(e^t - 1))$ $D_3(x) = (3/x^3) \int_0^x dt (t^3/(e^t - 1))$ $D_4(x) = (4/x^4) \int_0^x dt (t^4/(e^t - 1))$ $D_5(x) = (5/x^5) \int_0^x dt (t^5/(e^t - 1))$ $D_6(x) = (6/x^6) \int_0^x dt (t^6/(e^t - 1))$ $Li_2(x)$ $Li_2(x) = - \Re \int_0^x ds \log(1-s) / s$ $\Im(Li_2(x)) = 0$ $x <= 1$ $-\pi\log(x)$ $x > 1$ $z = r \exp(i \theta)$ $xy +/- xy \sqrt((dx/x)^2 +(dy/y)^2)$ $F(\phi,k)$ $E(\phi,k)$ $\Pi(\phi,k,n)$ $K(k) = F(\pi/2, k)$ $E(k) = E(\pi/2, k)$ $m = k^2$ $-n$ $RC(x,y)$ $RD(x,y,z)$ $RF(x,y,z)$ $RJ(x,y,z,p)$ $K(k)$ $E(k)$ $\Pi(k,n)$ $\sin^2(\alpha) = k^2$ $n \to -n$ $D(\phi,k)$ $sn(u\vert m)$ $cn(u\vert m)$ $dn(u\vert m)$ $erf(x)$ $erf(x) = (2/\sqrt(\pi)) \int_0^x dt \exp(-t^2)$ $erfc(x) = 1 - erf(x) = (2/\sqrt(\pi)) \int_x^\infty \exp(-t^2)$ $\log(\erfc(x))$ $Z(x) = (1/\sqrt@{2\pi@}) \exp(-x^2/2)$ $Q(x) = (1/\sqrt@{2\pi@}) \int_x^\infty dt \exp(-t^2/2)$ $h(x) \sim x$ $\exp(x)$ $y \exp(x)$ $(\exp(x)-1)/x$ $(\exp(x)-1)/x = 1 + x/2 + x^2/(2*3) + x^3/(2*3*4) + \dots$ $2(\exp(x)-1-x)/x^2$ $2(\exp(x)-1-x)/x^2 = 1 + x/3 + x^2/(3*4) + x^3/(3*4*5) + \dots$ $N$ $E_1(x)$ $E_2(x)$ $Ei(x)$ $PV$ $Shi(x) = \int_0^x dt \sinh(t)/t$ $Chi(x) := \Re[ \gamma_E + \log(x) + \int_0^x dt (\cosh[t]-1)/t]$ $\gamma_E$ $Ei_3(x) = \int_0^xdt \exp(-t^3)$ $Si(x) = \int_0^x dt \sin(t)/t$ $Ci(x) = -\int_x^\infty dt \cos(t)/t$ $AtanInt(x) = \int_0^x dt \arctan(t)/t$ $F_j(x)$ $F_@{-1@}(x) = e^x / (1 + e^x)$ $0$ $F_0(x) = \ln(1 + e^x)$ $F_1(x) = \int_0^\infty dt (t /(\exp(t-x)+1))$ $2$ $F_2(x) = (1/2) \int_0^\infty dt (t^2 /(\exp(t-x)+1))$ $j$ $F_j(x) = (1/\Gamma(j+1)) \int_0^\infty dt (t^j /(\exp(t-x)+1))$ $F_@{-1/2@}(x)$ $F_@{1/2@}(x)$ $F_@{3/2@}(x)$ $F_j(x,b)$ $F_0(x,b) = \ln(1 + e^@{b-x@}) - (b-x)$ $\Gamma(n)=(n-1)!$ $\Gamma(x)$ $\log(\Gamma(x))$ $\log(\vert\Gamma(x)\vert)$ $\Gamma(x) = sgn * \exp(resultlg)$ $\Gamma^*(x)$ $1/\Gamma(x)$ $\log(\Gamma(z))$ $z=z_r+i z_i$ $lnr = \log\vert\Gamma(z)\vert$ $arg = \arg(\Gamma(z))$ $(-\pi,\pi]$ $n! = \Gamma(n+1)$ $n!$ $n!! = n(n-2)(n-4) \dots$ $n!!$ $\log(n!)$ $\ln(\Gamma(n+1))$ $n < 170$ $\log(n!!)$ $= n!/(m!(n-m)!)$ $\log(n!) - \log(m!) - \log((n-m)!)$ $x^n / n!$ $n >= 0$ $(a)_x = \Gamma(a + x)/\Gamma(a)$ $a$ $a+x$ $(a,x)$ $\log((a)_x) = \log(\Gamma(a + x)/\Gamma(a))$ $a > 0$ $a+x > 0$ $result = \log(\vert(a)_x\vert)$ $sgn = \sgn((a)_x)$ $(a)_x = \Gamma(a + x)/\Gamma(a)$ $((a)_x - 1)/x$ $(a)_x = \Gamma(a + x)/\Gamma(a)$ $\Gamma(a,x) = \int_x^\infty dt t^@{a-1@} \exp(-t)$ $Q(a,x) = 1/\Gamma(a) \int_x^\infty dt t^@{a-1@} \exp(-t)$ $P(a,x) = 1 - Q(a,x) = 1/\Gamma(a) \int_0^x dt t^@{a-1@} \exp(-t)$ $P(a,x)$ $B(a,b) = \Gamma(a)\Gamma(b)/\Gamma(a+b)$ $b$ $\log(B(a,b))$ $I_x(a,b)=B_x(a,b)/B(a,b)$ $B_x(a,b) = \int_0^x t^@{a-1@} (1-t)^@{b-1@} dt$ $b > 0$ $0 <= x <= 1$ $C^@{(\lambda)@}_n(x)$ $n =1, 2, 3$ $\lambda > -1/2$ $n = 0, 1, 2, \dots, nmax$ $nmax >= 0$ $0F1(c,x)$ $1F1(m,n,x) = M(m,n,x)$ $1F1(a,b,x) = M(a,b,x)$ $U(m,n,x)$ $U(a,b,x)$ $2F1(a,b,c,x)$ $\vert x\vert < 1$ $(a,b,c,x)$ $x=1$ $c - a - b = m$ $2F1(a_R + i a_I, a_R - i a_I, c, x)$ $2F1(a,b,c,x) / \Gamma(c)$ $2F1(a_R + i a_I, a_R - i a_I, c, x) / \Gamma(c)$ $2F0(a,b,x)$ $x<0$ $2F0(a,b,x) = (-1/x)^a U(a,1+a-b,-1/x)$ $L^a_n(x) = ((a+1)_n / n!) 1F1(-n,a+1,x)$ $L_n(x)$ $L^0_n(x) = L_n(x)$ $L^k_n(x) = (-1)^k (d^k/dx^k) L_(n+k)(x)$ $L^a_1(x)$ $L^a_2(x)$ $L^a_3(x)$ $L^a_n(x)$ $a > -1$ $W(x)$ $W(x) \exp(W(x)) = x$ $W_0(x)$ $W > -1$ $W_@{-1@}(x)$ $W < -1$ $P_l(x)$ $l=1, 2, 3$ $\vert x\vert <= 1$ $dP_l(x)/dx$ $l = 0, \dots, lmax$ $Q_0(x)$ $x > -1$ $x != 1$ $Q_1(x)$ $Q_l(x)$ $P_l^m(x)$ $m$ $m >= 0$ $l >= m$ $dP_l^m(x)/dx$ $l = \vert m\vert, ..., lmax$

$$\sqrt@{(2l+1)/(4\pi)@} \sqrt@{(l-m)!/(l+m)!@} P_l^m(x)$$

$\vert x\vert <= 1.0$ $@var{lmax} - @var{m} + 1$ $P^\mu_@{-(1/2)+i\lambda@}(x)$ $Q^\mu_@{-(1/2)+i\lambda@}$ $P^@{1/2@}_@{-1/2 + i \lambda@}(x)$ $x > -1$ $P^@{-1/2@}_@{-1/2 + i \lambda@}(x)$ $P^0_@{-1/2 + i \lambda@}(x)$ $P^1_@{-1/2 + i \lambda@}(x)$ $P^@{-1/2-l@}_@{-1/2 + i \lambda@}(x)$ $l >= -1$ $P^@{-m@}_@{-1/2 + i \lambda@}(x)$ $m >= -1$ $H3d$ $\lambda \to \infty$ $\lambda\eta$ $L^@{H3d@}_0(\lambda,\eta) := \sin(\lambda\eta)/(\lambda\sinh(\eta))$ $\eta >= 0$ $L^@{H3d@}_0(\lambda,\eta) = j_0(\lambda\eta)$ $L^@{H3d@}_1(\lambda,\eta) := 1/\sqrt@{\lambda^2 + 1@} \sin(\lambda \eta)/(\lambda \sinh(\eta)) (\coth(\eta) - \lambda \cot(\lambda\eta))$ $L^@{H3d@}_1(\lambda,\eta) = j_1(\lambda\eta)$ $L^@{H3d@}_l(\lambda,\eta) = j_l(\lambda\eta)$ $L^@{H3d@}_l(\lambda, \eta)$ $0 <= l <= lmax$ $\log(x)$ $\log(\vert x\vert)$ $x \ne 0$ $z=z_r+i z_i$ $\exp(lnr + i \theta) = z_r + i z_i$ $\theta$ $[-\pi,\pi]$ $\log(1+x)$ $\log(1 + x) - x$ $ce_r(x,q)$ $se_r(x,q)$ $a=a_r(q)$ $a=b_r(q)$ $Mc^@{(j)@}_@{r@}(z,q)$ $Ms^@{(j)@}_@{r@}(z,q)$ $q$ $a_n(q)$ $b_n(q)$ $ce_n(q,x)$ $se_n(q,x)$ $ce(q,x)$ $se(q,x)$ $Mc_n^@{(j)@}(q,x)$ $Ms_n^@{(j)@}(q,x)$ $j = 3,4$ $M_n^@{(3)@} = M_n^@{(1)@} + iM_n^@{(2)@}$ $M_n^@{(4)@} = M_n^@{(1)@} - iM_n^@{(2)@}$ $M_n^@{(j)@} = Mc_n^@{(j)@}$ $Ms_n^@{(j)@}$ $\psi(x) = \Gamma'(x)/\Gamma(x)$ $\psi(n)$ $\psi(x)$ $1+i y$ $\Re[\psi(1 + i y)]$ $\psi'(n)$ $\psi'(x)$ $\psi^@{(n)@}(x)$ $x \int_x^\infty dt K_@{5/3@}(t)$ $x K_@{2/3@}(x)$ $J(n,x)$ $J(n,x) := \int_0^x dt t^n e^t /(e^t - 1)^2$ $J(2,x)$ $J(3,x)$ $J(4,x)$ $J(5,x)$ $\sin(x)$ $\cos(x)$ $\sinc(x) = \sin(\pi x) / (\pi x)$ $\sin(z_r + i z_i)$ $\cos(z_r + i z_i)$ $\log(\sin(z_r + i z_i))$ $\log(\sinh(x))$ $\log(\cosh(x))$ $x = r\cos(\theta)$ $y = r\sin(\theta)$ $x = r\cos(\theta)$ $[-\pi,\pi]$ $[0, 2\pi)$ $2\pi$ $\sin(x \pm dx)$ $\cos(x \pm dx)$ $\zeta(s) = \sum_@{k=1@}^\infty k^@{-s@}$ $\zeta(n)$ $n \ne 1$ $\zeta(s)$ $s \ne 1$ $\zeta(n) - 1$ $\zeta(s) - 1$ $\zeta(s,q) = \sum_0^\infty (k+q)^@{-s@}$ $\zeta(s,q)$ $s > 1$ $q > 0$ $\eta(s) = (1-2^@{1-s@}) \zeta(s)$ $\eta(n)$ $\eta(s)$ $J_0(5.0)$ $@var{n}-1$ $a'_i = a_i + b_i$ $a'_i = a_i - b_i$ $a'_i = a_i * b_i$ $a'_i = a_i / b_i$ $a'_i = x a_i$ $a'_i = a_i + x$ $(i,j)$ $@var{n1}-1$ $@var{n2}-1$ $m(i,j) = \delta(i,j)$ $k = 0$ $a'(i,j) = a(i,j) + b(i,j)$ $a'(i,j) = a(i,j) - b(i,j)$ $a'(i,j) = a(i,j) * b(i,j)$ $a'(i,j) = a(i,j) / b(i,j)$ $a'(i,j) = x a(i,j)$ $a'(i,j) = a(i,j) + x$ $p$ $p_i$ $v$ $v'$ $v'_i = v_@{p_i@}$ $(0,1,3,2)$ $(0,1,2,3)$ $v' = v P$ $(0,1,2,@dots{},n-1)$ $P$ $p_j$ $v' = v P^T$ $p = pa . pb$ $pb$ $c$ $c_i$ $(0,1,2,@dots{},k-1)$ $(n-k,n-k+1,@dots{},n-1)$ $@var{k}-1$ $@{0,1,2,3@}$ $O(N \log N)$ $O(kN)$ $O(N \log N)$ $y = \alpha x + y$ $y = \alpha A x + \beta y$ $C = \alpha A B + C$ $x^T y$ $\alpha x + y$ $A x$ $inv(A) x$ $A B$ $inv(A) B$ $\alpha + x^T y$ $x^H y$ $\vert\vert x\vert\vert _2 = \sqrt @{\sum x_i^2@}$ $\sum \vert x_i\vert$ $\sum \vert\Re(x_i)\vert + \vert\Im(x_i)\vert$ $\vert\Re(x_i)\vert + \vert\Im(x_i)\vert$ $(c,s)$ $(a,b)$ $(x', y') = (c x + s y, -s x + c y)$ $y = \alpha op(A) x + \beta y$ $op(A) = A$ $A^T$ $A^H$ $x = op(A) x$ $inv(op(A)) x$ $y = \alpha A x + \beta y$ $A = \alpha x y^T + A$ $A = \alpha x y^H + A$ $A = \alpha x x^T + A$ $A = \alpha x x^H + A$ $A = \alpha x y^T + \alpha y x^T + A$ $A = \alpha x y^H + \alpha^* y x^H A$ $C = \alpha op(A) op(B) + \beta C$ $C = \alpha A B + \beta C$ $C = \alpha B A + \beta C$ $B = \alpha op(A) B$ $B = \alpha B op(A)$ $B = \alpha op(inv(A))B$ $B = \alpha B op(inv(A))$ $C = \alpha A A^T + \beta C$ $C = \alpha A^T A + \beta C$ $C = \alpha A A^H + \beta C$ $C = \alpha A^H A + \beta C$ $C = \alpha A B^T + \alpha B A^T + \beta C$ $C = \alpha A^T B + \alpha B^T A + \beta C$ $C = \alpha A B^H + \alpha^* B A^H + \beta C$ $C = \alpha A^H B + \alpha^* B^H A + \beta C$ $A$ $LU$ $U$ $A x = b$ $L y = P b$ $U x = y$ $PA = LU$ $k = p_j$ $(-1)^n$ $r = A x - b$ $\ln\vert\det(A)\vert$ $\det(A)/\vert\det(A)\vert$ $M$ $QR$ $Q$ $Q^T Q = I$ $R$ $A x = b$ $R x = Q^T b$ $ran(A)$ $A = Q R$ $k=\min(M,N)$ $Q = Q_k ... Q_2 Q_1$ $Q_i = I - \tau_i v_i v_i^T$ $v_i$ $v_i = (0,...,1,A(i+1,i),A(i+2,i),...,A(m,i))$ $\vert\vert Ax - b\vert\vert$ $Q^T$ $Q^T v$ $Q v$ $R x = b$ $b' = Q^T b$ $w v^T$ $Q'R' = Q R + w v^T$ $Q'$ $R'$ $r$ $R y = Q^T b, x = P y$ $QRP^T$ $A = Q R P^T$ $A = Q R P^T$ $Q = Q_k ... Q_2 Q_1$ $R P^T x = Q^T b$ $Q'R' = Q R + w v^T$ $R P^T x = b$ $S$ $V$ $\sigma_i = S_@{ii@}$ $\sigma_1 >= \sigma_2 >= ... >= \sigma_N >= 0$ $A = U S V^T$ $M >= N$ $S_1$ $S_N$ $U S V^T$ $M>>N$ $\vert\vert A x - b\vert\vert _2$ $L^T$ $L y = b$ $L^T x = y$ $A = L L^T$ $T$ $Q T Q^T$ $U T U^T$ $H$ $H = U^T A U$ $N - 2$ $V' = VU$ $B$ $U B V^T$ $U^T U = I$ $\tau = 2/(v^T v)$ $P = I - \tau v v^T$ $\tau$ $P A$ $A P$ $P w$ $N >= 2$ $N >= 3$ $D$ $O(2n)$ $O(4n)$ $O(3n)$ $O(5n)$ $Z$ $Z = D Q$ $H(i,j) = 1/(i + j + 1)$ $V(x;i,j) = x_i^{n - j}$ $x = (-1,-2,3,4)$ $5.54555 + 3.08545i$ $0.9999984 + 0.0017674i$ $W\vec@{z@}$ $O(N^2)$ $W$ $f_1 f_2 ... f_n$ $O(N \sum f_i)$ $O(N \log_2 N)$ $x = FFT(z)$ $x = IFFT(z)$ $1/N$ $\Delta$ $-1/(2\Delta)$ $+1/(2\Delta)$ $N/2$ $+1/(2\Delta)$ $-1/(2\Delta)$ $-10$ $10$ $1/\sqrt N$ $t=128$ $k/N$ $2*2$ $2*3$ $O(n^2)$ $11*13$ $n=2*3*99991$ $N \sum f_i$ $f_i$ $=2*3*3*5*7$ $k=N/2$ $k < N/2$ $N-k$ $k > N/2$ $z_k = z^*_@{N-k@}$ $k = 0$ $n/2$ $z_k = z_@{N-k@}^*$ $n=5$ $n=6$ $w(x)$ $w(x)=1$ $(epsabs, epsrel)$ $RESULT$ $ABSERR = \vert RESULT - I\vert$ $2m+1$ $x_1, x_2, x_3$ $a < x_1 < x_2 < x_3 < b$ $(-\infty,+\infty)$ $(0,1]$ $x = (1-t)/t$ $(a,+\infty)$ $x = a + (1-t)/t$ $(-\infty,b)$ $x = b - (1-t)/t$ $x = c$ $(\alpha, \beta, \mu, \nu)$ $\alpha > -1$ $\beta > -1$ $\mu = 0, 1$ $\nu = 0, 1$ $\mu$ $f(x)$ $(x-a)^\alpha (b-x)^\beta \log^\mu (x-a) \log^\nu (b-x)$ $\sin(\omega x)$ $\cos(\omega x)$ $(\omega, L)$ $L = b - a$ $L/2^n$ $d$ $d\omega > 4$ $d\omega < 4$ $[a,+\infty)$ $\omega$ $\sin$ $\cos$ $c = (2 floor(\vert\omega\vert) + 1) \pi/\vert\omega\vert$ $C_k$ $u_k = (1 - p)p^@{k-1@}$ $p = 9/10$ $E_k$ $[0,n-1]$ $[0,2^32-1]$ $2^19937 - 1$ $10^6000$ $10^171$ $x_n$ $y_n$ $a_1 = 0$ $a_2 = 63308$ $a_3 = -183326$ $b_1 = 86098$ $b_2 = 0$ $b_3 = -539608$ $m_1 = 2^31 - 1 = 2147483647$ $m_2 = 2145483479$ $lcm(m_1^3-1, m_2^3-1)$ $2^185$ $10^56$ $a_1 = 107374182$ $a_2 = a_3 = a_4 = 0$ $a_5 = 104480$ $m = 2^31 - 1$ $10^46$ $2^32$ $^^$ $2^88$ $10^26$ $A=471$ $B=1586$ $C=6988$ $D=9689$ $2^D - 1$ $2^D$ $10^2917$ $a = 1103515245$ $c = 12345$ $m = 2^31$ $x_1$ $2^31$ $a = 25214903917$ $c = 11$ $m = 2^48$ $x_n/m$ $a = 44485709377909$ $2^48$ $2^46$ $2^250$ $2^800$ $a = 69069$ $c = 1$ $m = 2^32$ $a = 1664525$ $a = 65539$ $2^29$ $a = 16807$ $m = 2^31 - 1 = 2147483647$ $a_1 = 271828183$ $a_2 = 314159269$ $a = 1812433253$ $a = 62089911$ $a = 48271$ $a = 40692$ $m = 2^31 - 249$ $a = 1566083941$ $0 < x_i < 1$ $x_i$ $2^32-1 ~=~ 4.29e9$ $p(x)$ $x+dx$ $p dx$ $P(x)$ $Q(x)$ $P(x) + Q(x) = 1$ $0 <= P(x) <= 1$ $0 <= Q(x) <= 1$ $x=P^@{-1@}(P)$ $x=Q^@{-1@}(Q)$ $p(k)$ $\sum_k p(k) = 1$ $P(k)$ $Q(k)$ $P(k)+Q(k)=1$ $P(n)=1$ $Q(n)=0$ $P(1) = p(1)$ $Q(1)=1-p(1)$ $z = \mu + x$ $x > a$ $N(a;\sigma)$ $x,y$ $p(x,y)$ $-\infty < x < \infty$ $b = 1$ $b = 2$ $a = \sqrt@{2@} \sigma$ $\alpha = 1$ $\alpha = 2$ $\sigma = \sqrt@{2@} c$ $\alpha < 1$ $0 < alpha <= 2$ $[-1,1]$ $\tan(\pi \alpha/2)$ $-(2/\pi)\log\vert t\vert$ $\beta = 0$ $p(c, \alpha, \beta)$ $Y = X_1 + X_2 + \dots + X_N$ $p(N^(1/\alpha) c, \alpha, \beta)$ $a <= x < b$ $Y_i$ $Y_1$ $Y_2$ $\nu_1$ $\nu_2$ $F(x;\nu_1,\nu_2)$ $t(x;\nu)$ $-\infty < x < +\infty$ $x >= b$ $\vert v\vert^2 = x^2 + y^2 = 1$ $x=(u^2-v^2)/(u^2+v^2)$ $y=2uv/(u^2+v^2)$ $\vert v\vert^2 = x^2 + y^2 + z^2 = 1$ $v = (x_1,x_2,...,x_n)$ $\vert v\vert^2 = x_1^2 + x_2^2 + ... + x_n^2 = 1$ $0 < x < \infty$ $theta_i >= 0$ $alpha_i >= 0$ $\sum \theta_i = 1$ $a=alpha_i, b=1$ $p(\theta_1, ... , \theta_K)$ $K$ $P[k]$ $K+1$ $C[K]=1$ $u$ $C[k] <= u < C[k+1]$ $\log K$ $\lfloor uK \rfloor$ $O(K^2)$ $O(K)$ $k >= 0$ $0 <= k <= n$ $(n_1, n_2, ..., n_K)$ $sum_@{k=1@}^K n_k = N$ $(p_1, p_2, ..., p_K)$ $\sum p_i = 1$ $P(n_1, n_2, ..., n_K)$ $k >= 1$ $k=1$ $k-1$ $C(a,b) = a!/(b!(a-b)!)$ $t <= n_1 + n_2$ $max(0,t-n_2), ..., min(t,n_1)$ $n_1$ $n_2$ $t$ $x=2$ $\Hat\mu$ $\sigma^2 / N$ $\Hat\sigma^2$ $1/(N-1)$ $\sigma^2$ $2 \sigma^4 / N$ $w_i$ $\sigma_i^2$ $w_i = 1/\sigma_i^2$ $x_i >= x_j$ $x_i <= x_j$ $(n-1)/2$ $(n - 1)f$ $\delta$ $(n-1)f - i$ $@var{n}+1$ $n+1$ $r <= x$ $x < r$ $d = (xmax-xmin)/n$ $h'_1(i) = h_1(i) + h_2(i)$ $h'_1(i) = h_1(i) - h_2(i)$ $h'_1(i) = h_1(i) * h_2(i)$ $h'_1(i) = h_1(i) / h_2(i)$ $h'_1(i) = h_1(i) * scale$ $h'_1(i) = h_1(i) + offset$ $p(x)dx$ $n_i$ $s$ $sum[i] <= r < sum[i+1]$ $\delta$ $(r - sum[i])/(sum[i+1] - sum[i])$ $(x,y)$ $n(x,y)$ $E$ $n(x,E)$ $@var{nx} + 1$ $@var{ny} + 1$ $h'_1(i,j) = h_1(i,j) + h_2(i,j)$ $h'_1(i,j) = h_1(i,j) - h_2(i,j)$ $h'_1(i,j) = h_1(i,j) * h_2(i,j)$ $h'_1(i,j) = h_1(i,j) / h_2(i,j)$ $h'_1(i,j) = h_1(i,j) scale$ $h'_1(i,j) = h_1(i,j) + offset$ $p(x,y) dx dy$ $n_@{ij@}$ $A_@{ij@}$ $(x,y,z)$ $E^2=x^2+y^2+z^2$ $E^2$ $((x_l,x_u)$ $(y_l,y_u), ...)$ $f(x,params)$ $a = 3$ $E(f; N)$ $\sigma(E;N)$ $\Var(f)/N$ $\Var(f)$ $\sigma(f)/\sqrt@{N@}$ $1/\sqrt@{N@}$ $E_a(f)$ $E_b(f)$ $\sigma_a^2(f)$ $\sigma_b^2(f)$ $E(f) = (1/2) (E_a(f) + E_b(f))$ $N_a$ $N_b$ $\alpha$ $\vert f\vert$ $g$ $E_g(f; N)$ $g = \vert f\vert/I(\vert f\vert)$ $V_g(f; N)$ $K^d$ $g(x_1, x_2, ...) = g_1(x_1) g_2(x_2) ...$ $Kd$ $I = \Gamma(1/4)^4/(4 \pi^3) = 1.393203929685676859...$ $(0,0,0)$ $(\pi,\pi,\pi)$ $(0,\pi,\pi)$ $(\pi,0,\pi)$ $(\pi,\pi,0)$ $E_@{i+1@} > E_i$ $p = 1$ $E_@{i+1@} <= E_i$ $E_@{i+1@} - E_i$ $T -> T/mu_T$ $\mu_T$ $i = 1, \dots, n$ $J_@{ij@} = df_i(t,y(t)) / dy_j$ $f_i(t,y,params)$ $df_i(t,y,params)/dt$ $J_@{ij@}$ $t+h$ $h$ $@var{h}/10$ $y(t)$ $y'(t)$ $D_i$ $E_i = \vert yerr_i\vert$ $q=4$ $E/D$ $E_i/D_i$ $1/5$ $y_i(t)$ $y'_i(t)$ $s_i$ $(t_0, t_1)$ $y = x'(t)$ $(y, y') = (1, 0)$ $t=0$ $t=100$ $10^@{-6@}$ $t = 0, 1, 2, \dots, 100$ $(x_1, y_1) \dots (x_n, y_n)$ $y(x)$ $y(x_i) = y_i$ $(x_i, y_i)$ $x_i = i + \sin(i)/2$ $y_i = i + \cos(i^2)$ $i = 0 \dots 9$ $x-h$ $x-h/2$ $x+h/2$ $x+h$ $x+h/4$ $x+3h/4$ $f(x) = x^@{3/2@}$ $x=0$ $f(x) = \sum c_n T_n(x)$ $T_n(x) = \cos(n \arccos x)$ $1 / \sqrt@{1-x^2@}$ $T_0(x) = 1$ $T_1(x) = x$ $T_2(x) = 2 x^2 - 1$ $[a,b]$ $c[0]$ $O(N)$ $O(2n^2 + 3n)$ $\zeta(2) = \pi^2 / 6$ $O(1/N)$ $10^10$ $\psi_@{s,\tau@}$ $2^j$ $@{\psi_@{j,n@}@}$ $k/2$ $k=4, 6, @dots{}, 20$ $k=2$ $k = 100*i + j$ $f_i -> w_@{j,k@}$ $j = 0 ... J-1$ $k = 0 ... (2^j)-1$ $J = \log_2(n)$ $s_@{-1,0@}$ $d_@{j,k@}$ $^2$ $f(t)$ $g_m$ $g_m=0$ $m > M$ $M-1$ $[0,X]$ $(j_@{\nu,n+1@}/j_@{\nu,M@}) X$ $j_@{\nu,n+1@}/X$ $f(x) = (x-x_0)^2$ $f(x) = (x-x_0)^3$ $f'(x,params)$ $f'(x)$ $f(x) = 2\exp(2x)$ $y = f(x)$ $dy = f'(x)$ $x = [a,b]$ $\min(\vert a\vert,\vert b\vert)$ $\vert x\vert$ $r^*$ $\vert f(x)\vert$ $f(a)$ $f(b)$ $(a,f(a))$ $(b,f(b))$ $(1 + \sqrt 5)/2$ $1.62$ $R_i$ $x = \sqrt 5 = 2.236068...$ $x'$ $f(x') < f(x)$ $f(a) > f(x) < f(b)$ $x^*$ $\sqrt \epsilon$ $\epsilon$ $x^4$ $x_m$ $x_m^*$ $f(a) > f(x) < f(b)$ $(3-\sqrt 5)/2 = 0.3189660$ $f(a') > f(x') < f(b')$ $f(x) = \cos(x) + 1$ $x = \pi$ $(0,6)$ $J$ $J_@{ij@} = d f_i / d x_j$ $n^2$ $J^@{-1@}$ $A = 10^4$ $J_ij = d f_i(x,params) / d x_j$ $J(x)$ $dx$ $\vert D (x' - x)\vert < \delta$ $J dx = - f$ $\vert x' - x\vert < \delta$ $F$ $\vert f(x')\vert^2/\vert f(x)\vert^2$ $\delta_j$ $\sqrt\epsilon \vert x_j\vert$ $\epsilon \approx 2.22 \times 10^-16$ $df$ $a = 1, b = 10$ $(x,y) = (1,1)$ $(-10,-5)$ $(-4.23,-65.3)$ $g = \nabla f$ $dot(p,g) < tol \vert p\vert \vert g\vert$ $g_i = d f(x,params) / d x_i$ $g(x)$ $\delta f = g \delta x$ $p' = g' - \beta g$ $\beta=-\vert g'\vert^2/\vert g\vert^2$ $\beta$ $\sigma$ $\chi^2$ $Y(c,x)$ $c = @{c_0, c_1, @dots{}@}$ $w_i = 1/\sigma_i^2$ $\sigma_i$ $y_i$ $p \times p$ $C_@{ab@} = <\delta c_a \delta c_b>$ $< >$ $\delta c_a$ $\delta y_i$ $<\delta y_i \delta y_j> = \sigma_i^2 \delta_@{ij@}$ $w = 1/\sigma^2$ $\sigma^2 = \sum (y_i - Y(c,x_i))^2 / (n-p)$ $\sigma_@{c_a@} = \sqrt@{C_@{aa@}@}$ $Y(c,x) = c_0 + c_1 x$ $Y = c_0 + c_1 X$ $Y = c_0 + c_1 X$ $Y = c_1 X$ $Y = c_1 X$ $y = X c$ $X$ $\chi^2 = \sum_i w_i (y_i - \sum_j X_@{ij@} c_j)^2$ $p-1$ $\omega_1$ $\omega_2$ $\omega_p$ $x_2$ $x_p$ $x_j(i)$ $x_j$ $s_i/s_0$ $y = x.c$ $y = c_0 + c_1 x + c_2 x^2$ $c_0$ $c_1 x$ $c_2 x^2$ $y = e^x$ $0 < x < 2$ $e^x$ $O(x^3)$ $\vert\vert F\vert\vert$ $Y(x,t)$ $t_i$ $J_@{ij@} =(1 / \sigma_i) d Y_i / d x_j$ $Y_i = Y(x,t_i)$ $\Phi(x) = (1/2) \vert\vert F(x)\vert\vert^2$ $g = J^T f$ $\vert F + J p\vert$ $\vert D p\vert < \Delta$ $f_i = (Y(x, t_i) - y_i) / \sigma_i$ $\delta f = J \delta c$ $<\delta f \delta f^T> = I$ $f_i = (Y(x, t_i) - y_i)$ $\sigma^2 = \sum (y_i - Y(x,t_i))^2 / (n-p)$ $\sigma^2 C$ $Y = A \exp(-\lambda t) + b$ $t_i = i$ $\lambda$ $x_0 = A$ $x_1 = \lambda$ $x_2 = b$ $chi^2/dof >> 1$ $\sqrt@{\chi^2/dof@}$ $t = \{t_0, t_1, \dots, t_{n+k-1}\}$ $i = 0, \dots, n-1$ $k = 4$ $(x_j, f(x_j))$ $n = nbreak + k - 2$ $O(5k + nbreak)$ $B_i(x)$ $y(x) = \cos{(x)} \exp{(-0.1 x)}$ $[0, 15]$ $G$ $\mu_0$ $\epsilon_0$ $\hbar$ $R_0$ $V_0$ $au$ $ly$ $pc$ $eV$ $amu$ $m_e$ $m_\mu$ $m_p$ $m_n$ $Ry$ $Ry = h c R$ $a_0$ $\mu_B$ $\mu_N$ $\mu_e$ $\mu_p$ $\sigma_T$ $btu$ $10^-5$ $10^24$ $10^21$ $10^18$ $10^15$ $10^12$ $10^9$ $10^6$ $10^3$ $10^-3$ $10^-6$ $10^-9$ $10^-12$ $10^-15$ $10^-18$ $10^-21$ $10^-24$ $(s,E,f)$ $E_min$ $E_max$ $e = E + bias$ $fffff...$ $2^(E_min)$ $2^(E_min - p)$ $2^(E_min - 1)$ $2^(E_max + 1)$ $1/3$ $0.01010101...$ $4 \times 10^-16$ $4 \times 10^-15$ $O(10^-7)$