complex3 Mackichan (S. Swanson) around 300 equation tests

\begin{matrix} 2 a b x^{3} f^{} (x) + \sin \cos = 1 f (z) =_{n = 0}^{} a_{n} z^{n}, | z | < R (R 0)_{C} (_{n = 0}^{} a_{n} z^{n}) z =_{n = 0}^{} a_{n}_{C} z^{n} z lim_{n} |_{C} [f (z)_{k = 0}^{n} a_{k} z^{k}] z | = 0 n N () | f (z)_{k = 0}^{n} a_{k} z^{k} | < 10 Bq + 10 Ci 10 amol + 10 Emol 10 fmol + 10 Gmol 10 kmol + 10 Mmol 10 mol + 10 mmol 10 mol + 10 nmol 10 Pmol + 10 pmol 10 Tmol 10 acre + 10 hectare 10 {ft}^{2} + 10 {in}^{2} 10 m^{2} 10 A + 10 kA 10 A + 10 mA 10 nA 10 F + 10 F 10 mF + 10 nF 10 pF 10 C + 1.0 m/s/s 0.1 m / s^{2} 10 kS + 10 S 10 mS + 10 S 10 kV + 10 MV 10 V + 10 mV 10 nV + 10 pV 10 V 10 G + 10 k 10 M + 10 m 10 10 Btu + 10 cal 10 eV + 10 erg 10 GeV + 10 GJ 10 J + 10 kcal 10 kJ + 10 MeV 10 MJ + 10 J 10 mJ + 10 nJ 10 dyn + 10 kN 10 MN + 10 N 10 mN + 10 N 10 ozf + 10 lbf 10 EHz + 10 GHz 10 Hz + 10 kHz 10 MHz + 10 PHz 10 THz 10 fc + 10 lx 10 phot 10 + 10 am 10 cm + 10 dm 10 fm + 10 ft 10 in 10 km + 10 m 10 m + 10 mi 10 mm + 10 nm 10 pm 10 sb 10 lm 10 cd 10 Mx + 10 Wb 10 mWb + 10 nWb 10 Wb 10 G + 10 T 10 mT + 10 nT 10 pT + 10 T 10 H + 10 H 10 mH 10 u + 10 cg 10 dg + 10 g 10 kg + 10 g 10 mg + 10 lb 10 slug 10 + 10 rad 10 mrad + 10^{} 10 rad + 10^{} 10 GW + 10 hp 10 kW + 10 MW 10 W + 10 mW 10 nW + 10 W 10 atm + 10 bar 10 kbar + 10 kPa 10 MPa + 10 Pa 10 mbar + 10 mmHg 10 Pa + 10 torr 10 sr 10 C + 10 F 10 K 10 as + 10 d 10 fs + 10 h 10 s + 10 ms 10 min + 10 ns 10 ps + 10 s 10 y 10 {ft}^{3} + 10 {in}^{3} 10 m^{3} + 10 gal 10 l 10 ml + 10 pint 10 qt \frac{1}{x (y)} = (e^{\frac{1}{2} y^{2}} \sin y y + C_{1}) e^{\frac{1}{2} y^{2}}_{x} y y = \sin x (\binom{1}{2}) (\binom{1}{2}) (\binom{1}{2}) [\binom{1}{2}] (\binom{1}{2}) {\binom{1}{2}} \frac{1}{2} \frac{1}{2} \frac{1}{2} \frac{1}{2} \frac{1}{2} \frac{1}{2} \frac{1}{2} \frac{1}{2} \frac{1}{2} \frac{1}{2} \frac{1}{2} \frac{1}{2} \binom{1}{2} \binom{1}{2} \binom{1}{2} (a b) = b a \frac{2}{5} + \frac{3}{7} = \frac{2 7 + 3 5}{35} = \frac{29}{35} | a | = {\begin{aligned} a & if & a 0 \\ a & if & a < 0 \end{aligned} a^{n} = \underset{n factors}{\underset{}{a a a}} {(\frac{a}{b})}^{n} = {(\frac{b}{a})}^{n} \sqrt[n]{a} = b means b^{n} = a . \sqrt[4]{\frac{16}{81}} = \frac{\sqrt[4]{16}}{\sqrt[4]{81}} = \frac{2}{3} {x x 0, x 1} a_{n} x^{n} + a_{n 1} x^{n 1} + + a_{1} x + a_{0} a^{3} b^{3} = (a b) (a^{2} + a b + b^{2}) {(x + y)}^{2} H = {(\begin{matrix} a & b \\ c & d \end{matrix}) G a d b c = 1} | x | + || y || + {z} [a c] + (b) = [a, b] x = 1 x = 1 x = 1 x = 1 [\frac{10}{3}, \frac{7}{3}) (\frac{7}{3}, \frac{4}{3}] A \frac{u}{x} + B \frac{u}{y} + C u = E_{}^{} x_{\begin{matrix} 1 < i < 10 \\ 1 < j < 10 \end{matrix}}^{} 2^{i + j}_{1_{}^{}}^{1^{\begin{matrix} 5^{\begin{matrix} 7 \\ 6 \end{matrix}} \\ 2^{\begin{matrix} 4 \\ 3 \end{matrix}} \end{matrix}}} y (x) = \frac{x e^{x} e^{x} + 2}{e^{x}} = x 1 + \frac{2}{e^{x}} \begin{array}{r} _{x x} y y = 0 \\ y (0) = 1 \\ y^{} (0) = 0 \end{array} y (x) = \frac{1}{3} e^{\sqrt[3]{(1)} x} + \frac{2}{3} e^{\frac{1}{2} \sqrt[3]{(1)} x} \cos \frac{1}{2} \sqrt{3} \sqrt[3]{(1)} x y (t) = 2 \tan (2 t \frac{1}{4}) (\begin{matrix} e^{2 i x} \\ 2 Dirac (x 2) \end{matrix}, x, s) = (\begin{matrix} 2 Dirac (s 2) \\ 2 e^{2 i s} \end{matrix}) \begin{matrix} x = 1 \\ x + 3 = 123 \end{matrix} \begin{array}{rrrr} t & x & y & z \\ 0 & 1.0000 & 1.0000 & 1.0000 \\ .1 & 1.1158 & 1.0938 & .8842 \\ .2 & 1.2668 & 1.1695 & .7332 \\ .3 & 1.4582 & 1.2173 & .5418 \\ .4 & 1.6953 & 1.2253 & .3047 \\ .5 & 1.9830 & 1.1791 & .0170 \\ .6 & 2.3256 & 1.0619 & .3256 \\ .7 & 2.7265 & .8542 & .7265 \\ .8 & 3.1873 & .5344 & 1.1873 \\ .9 & 3.7077 & .0777 & 1.7077 \\ 1.0 & 4.2842 & .5424 & 2.2842 \end{array} K_{v} (z) = {BesselK}_{v} (z) z^{2} \frac{^{2} w}{z^{2}} + z \frac{w}{z} (z^{2} + v^{2}) w = 0 \frac{^{2} u (x, y)}{x^{2}} \frac{^{2} u (x, y)}{y^{2}} = 0 y (t, x) = F_{1} (x a t) + F_{2} (x a t) \begin{array}{lcr} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array} 2 x + 1 = 5 \begin{matrix} 1 = 3 \\ 9 = 7 \end{matrix} \begin{matrix} a b \\ c d \\ e f \end{matrix} \begin{matrix} x + 2 y 3 = 5 \\ 4 x y 5 = 98 \end{matrix} \begin{matrix} x = z \\ 1 = 3 \end{matrix} \begin{matrix} A_{1} = N_{0} (;^{}) (;^{}), \\ A_{2} = (;^{}) (;), \\ A_{3} = N (;) . \end{matrix} \begin{matrix} \sin \\ \cos \end{matrix} x = {\begin{cases} x & if x < 0 \\ x & if x 0 \end{cases} \begin{matrix} L M R M \\ L M R M \end{matrix} \begin{matrix} M A T H \\ M A T H \end{matrix} F = 0 F F =^{2} F + 7 = A (x y, y z, z x) = [\begin{matrix} y \\ z \\ x \end{matrix}] (y, z, x) = (1, 1, 1) 0 x + y + = 102 a + b = c x + 1 x + f (x) 1 = 123 T h e q u i c k b r o w n f o x j u m p s o v e r t h e l a z y d o g . T h e e n d . (\frac{f}{x_{1}} (c_{1}, c_{2},, c_{n}), \frac{f}{x_{2}} (c_{1}, c_{2},, c_{n}),, \frac{f}{x_{n 1}} (c_{1}, c_{2},, c_{n})) (c u v + v^{2} w) = (u v, c v, c u + 2 v w, v^{2}) \begin{matrix} D_{u} f (a, b, c) = f (a, b, c) u \\ = \frac{f}{x} (a, b, c) u_{1} + \frac{f}{y} (a, b, c) u_{2} + \frac{f}{z} (a, b, c) u_{3} \end{matrix} {+ 2 X_{3} (\arccos \frac{1}{7} \sqrt{14}) | X_{3}}, {2 X_{4} + (\arccos \frac{1}{7} \sqrt{14}) | X_{4}} P = A {(A^{T} A)}^{1} A^{T} det (\begin{matrix} x & y & 1 \\ a & b & 1 \\ a & d & 1 \end{matrix}) = x b x d + a d a b = 0 A () A () = [\begin{array}{rr} \cos & \sin \\ \sin & \cos \end{array}] [\begin{array}{rr} \cos & \sin \\ \sin & \cos \end{array}] J (A) = [\begin{matrix} J_{n_{1}} (_{1}) & 0 & 0 \\ 0 & J_{n_{2}} (_{2}) & 0 \\ 0 & 0 & J_{n_{k}} (_{k}) \end{matrix}] det (\begin{matrix} 4 + X & 1 & 0 \\ 0 & 4 + X & 0 \\ 0 & 0 & 4 + X \end{matrix}) = {(X 4)}^{3} {(\begin{matrix} \frac{1}{2} \frac{1}{6} \sqrt{33} \\ 1 \end{matrix})} \frac{5}{2} \frac{1}{2} \sqrt{33} A = max_{x 0} \frac{A x}{x} (\begin{matrix} a_{1 1} & a_{1 2} \\ a_{2 1} & a_{2 2} \end{matrix}) + (\begin{matrix} b_{1 1} & b_{1 2} \\ b_{2 1} & b_{2 2} \end{matrix}) = (\begin{matrix} a_{1 1} + b_{1 1} & a_{1 2} + b_{1 2} \\ a_{2 1} + b_{2 1} & a_{2 2} + b_{2 2} \end{matrix}) f ([\begin{matrix} 1 & 2 \\ 4 & 3 \end{matrix}]) = {[\begin{matrix} 1 & 2 \\ 4 & 3 \end{matrix}]}^{2} 5 [\begin{matrix} 1 & 2 \\ 4 & 3 \end{matrix}] 2 = [\begin{matrix} 2 & 2 \\ 4 & 0 \end{matrix}] x = lim_{x = 1}_{1}^{2} a_{a}^{b} f (x) x = lim_{P 0}_{i = 1}^{n} f ({\overset{}{x}}_{i}) x_{i}_{a}^{b} f (x) x = lim_{n} \frac{b a}{n}_{i = 1}^{n} f (a + i \frac{b a}{n})_{0}^{2} x^{5} \sqrt{x^{3} + 1} x =_{1}^{3} \frac{2}{3} u \frac{\sqrt{(u^{2})}}{{(u^{2} 1)}^{\frac{2}{3}}} (u^{2} {(u^{2} 1)}^{\frac{2}{3}} {(u^{2} 1)}^{\frac{2}{3}}) u f (g (x)) g^{} (x) x = f (u) u x = 2_{n = 1}^{100} n (n 1) lim_{x 0} \sin (\frac{1}{x}) = 1 .. 1 h (i, j) = (2 j) g (i) + (j 1) f (g (i)) : [0, 1] [0, 1] 0 x = x x y = h^{1} (h (x) h (y)) x y = f^{1} (max {f (x) + f (y) 1, 0}) x y = ((x) (y)) x_{0} y = {\begin{matrix} x y & if & x y = 1 \\ 0 & if & x y < 1 \end{matrix} lim_{a 1^{+}} \log_{a} [1 + \frac{(a^{x} 1) (a^{y} 1)}{a 1}] = lim_{a 1^{}} \log_{a} [1 + \frac{(a^{x} 1) (a^{y} 1)}{a 1}] = x y g (x) = \exp (\frac{1 {(1 x)}^{a}}{(2^{a} 1) {(1 x)}^{a}}) Aut (I) = {f : [0, 1] [0, 1] x^{2} + y^{2} = r^{2}, \tan = \frac{y}{x} \sqrt{2} \sqrt{1 t^{2}} [(2 + \sin t) 10 \cos t, (2 + \cos t) 10 \sin t, 3 \sin 3 t] {t = 0, s = 0}, {t =, s =} \begin{matrix} \begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 2 & 4 & 6 & 8 & 10 & 1 & 3 & 5 & 7 & 9 \\ 3 & 6 & 9 & 1 & 4 & 7 & 10 & 2 & 5 & 8 \\ 4 & 8 & 1 & 5 & 9 & 2 & 6 & 10 & 3 & 7 \\ 5 & 10 & 4 & 9 & 3 & 8 & 2 & 7 & 1 & 6 \\ 6 & 1 & 7 & 2 & 8 & 3 & 9 & 4 & 10 & 5 \\ 7 & 3 & 10 & 6 & 2 & 9 & 5 & 1 & 8 & 4 \\ 8 & 5 & 2 & 10 & 7 & 4 & 1 & 9 & 6 & 3 \\ 9 & 7 & 5 & 3 & 1 & 10 & 8 & 6 & 4 & 2 \\ 10 & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 \end{matrix} \end{matrix} \begin{matrix} testing x^{2} end. \end{matrix} \begin{matrix} x \end{matrix} \begin{matrix} x \end{matrix} \begin{matrix} x \end{matrix} \begin{matrix} x \end{matrix} \frac{f}{x} (x_{1}) = 5 x x = x y x y = x y z x y z = x y z t x y z t mod a 5 mod 3 = 2 f (0) mod 3 = 1 5 x + 4 8 (mod 13) a = (5 3) / 5 mod 7 = 6 (2 x^{2} + x + 2) + (2 x + 1) mod 3 = 2 x^{2} \begin{array}{rcl} + & 0 & 1 \\ 000 & 000 & 111 \\ 1 & 1 & 0 \end{array} 4.9749 \frac{}{x} F (x) [86.333, 146.33, 129.33] BinomialDist (x; n, p) =_{k = 0}^{x} (\binom{n}{k}) p^{k} q^{n k} Pr (X 54) = BinomialDist (54; 100, .55) = .45846 k = max {| \frac{f}{y} (x, y) | : (x, y) D} . m = lim_{x \overset{}{} a} \frac{f (x) f (a)}{x a} | A | = | \begin{array}{llllll} a_{1 1} & a_{1 2} & a_{1 n} \\ a_{2 1} & a_{2 2} & a_{2 n} \\ a_{n 1} & a_{n 2} & a_{n n} \end{array} | = a_{1 1} A_{1 1} + a_{1 2} A_{1 2} + + a_{1 n} A_{1 n} x = 1 (hl text x end.) x = 1 (hl to URI x end) x = 1 (sex) x = 1 (jbm) f (x) g [y] h {z} + a b c \frac{123}{\frac{456}{A}} | \frac{A}{\frac{B}{A}} / \frac{1}{\frac{2}{A}} / (\frac{3}{\frac{4}{A}}) \frac{5}{\frac{6}{A}} \frac{7}{\frac{8}{A}} \frac{\frac{9}{20}}{\frac{10}{A}} \frac{11}{\frac{12}{A}} \frac{13}{\frac{14}{A}} \frac{15}{\frac{16}{A}} \frac{17}{\frac{18}{A}} x \begin{array}{ll} x & x \\ x & x \end{array} x (a_{1}, a_{2},, a_{n}) (b_{1}, b_{2},, b_{n}) = a_{1} b_{1}^{} + a_{2} b_{2}^{} + + a_{n} b_{n}^{} \frac{n}{5} + \frac{n}{5^{2}} + \frac{n}{5^{3}} + \frac{n}{5^{4}} + x_{1} + + x_{n} \underset{k times}{\underset{}{x + + x}} \sqrt[n]{x_{1} x_{2} x_{n}} n! = 1 2 3 4 n P : a = x_{0} < x_{1} < x_{2} < < x_{n} = b f (x) = \frac{30}{13 \cos x} + \frac{10}{3} \sqrt{(100 + \frac{9}{\cos^{2} x} \frac{60}{\cos x} \sin (x + \frac{29}{90}))} \cos (A x) \sin (B x) x = \frac{\cos (B A) x}{2 (B A)} + \frac{\cos (B + A) x}{2 (B + A)} + C . 235.3 + 813 = 1048.3 max_{2 x 2} (x^{3} 6 x + 3) = 8.0 x decade = 2 century \frac{^{5} (x^{7} 3 x^{6})}{x^{5}} \frac{^{n} \sin x}{x^{n}} \frac{^{3}}{x^{3}} f (x) \frac{^{2}}{t^{2}} (4 t^{5} 3 t) f (x) = \frac{30}{13 \cos x} + \frac{10}{3} \sqrt{(100 + \frac{9}{\cos^{2} x} \frac{60}{\cos x} \sin (x + \frac{29}{90}))}_{R^{3}} (\frac{{| u_{1} |}^{2} + {| u_{0} |}^{2}}{2} + \frac{{| u_{0} |}^{6}}{6}) x < (F) k = z + 1 M \frac{M^{\frac{M}{M}}}{M}_{x} x^{2}_{x} (x^{2})_{x x} (x^{2})_{x^{2}} (x^{2})_{x y} (x^{2} y^{3})_{x^{s} y^{t}} (x^{2} y^{3}) 5 \begin{matrix} \begin{matrix} x + \sqrt[2]{\frac{a^{y 1}}{12.34}} & \sin \\ 1 \end{matrix} \end{matrix} \begin{array}{c} 0 & 1 \\ 1 & 0 \end{array} (\begin{array}{c} 0 & i \\ i & 0 \end{array}) [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}] | \begin{array}{c} a & b \\ c & d \end{array} | \begin{array}{c} 1 & 0 & 1 \\ 0 & 11 \end{array} \begin{array}{c} 1 & 2 & 3 \\ 4 & 5 \end{array} testing \begin{matrix} \sin \end{matrix} \overset{}{a} + \overset{}{b} + \overset{}{c} + \overset{}{d} + \overset{}{e} + \overset{}{f} + \overset{}{g} + h + \overset{}{i} + \overset{}{j} + \overset{}{k} + \overset{}{l} + \overset{}{m} + \overset{}{n} f (g (x)) = \sin^{3} x^{2} + \sin x^{2} \sin (\sin x^{2}) (x^{2} + 12 x^{2} + 12) + 1234 \begin{array}{llll} x = 1 & not & here \\ x^{2} & merged & y_{1} \\ jbm & lowlife & The end. \end{array} x^{2} + y^{2} = z^{2} 1 \begin{matrix} x^{2} + y^{2} = z^{2} 1 \\ x + y^{3} = z^{3} \end{matrix} \begin{matrix} x^{2} + y^{2} = z^{2} 1 \\ x + y^{3} = z^{3} \end{matrix} \begin{matrix} x^{2} + y^{2} = 1 \\ x = \sqrt{1 y^{2}} \end{matrix} \begin{matrix} {(a + b)}^{2} = a^{2} + 2 a b + b^{2} \\ (a + b) (a b) = a^{2} b^{2} \end{matrix} \begin{matrix} First line of equation \\ Middle line of equation \\ Other middle line of equation \\ Last line of equation \end{matrix} \begin{matrix} L_{1} = R_{1} L_{2} = R_{2} \\ L_{3} = R_{3} L_{4} = R_{4} \end{matrix} \begin{matrix} {(a + b)}^{4} = {(a + b)}^{2} {(a + b)}^{2} \\ = (a^{2} + 2 a b + b^{2}) (a^{2} + 2 a b + b^{2}) \\ = a^{4} + 4 a^{3} b + 6 a^{2} b^{2} + 4 a b^{3} + b^{4} \end{matrix} \begin{matrix} x^{2} + y^{2} = 1 \\ x = \sqrt{1 y^{2}} \end{matrix} \begin{matrix} {(a + b)}^{2} = a^{2} + 2 a b + b^{2} \\ (a + b) (a b) = a^{2} b^{2} \end{matrix} \begin{array}{ll} Vertex & V (0, 0) \\ Focus & F (0, p) \\ Directrix & y = p \end{array} \frac{}{x} (\csc^{1} x) = \frac{1}{| x | \sqrt{x^{2} 1}} \tanh^{1} x = \frac{1}{2} \ln (\frac{1 + x}{1 x}) 1 < x < 1 + A B C + 1 = a b c y = e^{P x} [e^{P x} Q x + c] x = 1 + y^{3} & x = 1 + y $ 1.00 + 25 3 + 2.45 0.7 a ₠ + 20 ₣ + 30 ₤ 4.56 ₧ \begin{matrix} 2 x + y = 3 \\ 3 x 4 y = 5 \\ a + b = c + 12345 \end{matrix} \begin{matrix} Unrestricted & Symmetric & Antisymmetric & Triangular \end{matrix} a b x c d y e f 11 g h Z k l 3 A B C A B C 10 11 12 x y z \lim^{} x \lim_{} x \lim_{} x \lim_{} x \begin{matrix} x = y + z \\ = k + m \end{matrix} \begin{array}{l} College Algebra Second Edition \\ James Stewart McMaster Universitiy \\ Lothar Redlin Pennsylvania State University \\ Saleem Watson California State University, Long Beach \\ Copyright 1996, ISBN 0 534-33983-2 \\ Brooks/Cole Publishing Company \\ An International Thomson Publishing Company \end{array} {\frac{\frac{1}{2}}{\frac{1}{2}}_{1}^{2}} \frac{\frac{1}{2}}{\frac{1}{2}} |_{1}^{2} \frac{\frac{1}{2}}{\frac{1}{2}} |_{1}^{2} \frac{\frac{1}{2}}{\frac{1}{2}}_{1}^{2} [\frac{\frac{1}{2}}{\frac{1}{2}}] (\frac{\frac{1}{2}}{\frac{1}{2}}) {\frac{\frac{1}{2}}{\frac{1}{2}}} \frac{\frac{1}{2}}{\frac{1}{2}} \frac{\frac{1}{2}}{\frac{1}{2}} \frac{\frac{1}{2}}{\frac{1}{2}} \frac{\frac{1}{2}}{\frac{1}{2}} \frac{\frac{1}{2}}{\frac{1}{2}} \frac{\frac{1}{2}}{\frac{1}{2}} \frac{\frac{1}{2}}{\frac{1}{2}} \frac{\frac{1}{2}}{\frac{1}{2}} \frac{\frac{1}{2}}{\frac{1}{2}} \frac{\frac{1}{2}}{\frac{1}{2}} \arrowvert \frac{\frac{1}{2}}{\frac{1}{2}} \arrowvert \Arrowvert \frac{\frac{1}{2}}{\frac{1}{2}} \Arrowvert \bracevert \frac{\frac{1}{2}}{\frac{1}{2}} \bracevert | \frac{\frac{1}{2}}{\frac{1}{2}} | | \frac{\frac{1}{2}}{\frac{1}{2}} | | \frac{\frac{1}{2}}{\frac{1}{2}} | \frac{\frac{1}{2}}{\frac{1}{2}} \frac{\frac{1}{2}}{\frac{1}{2}} / \frac{\frac{1}{2}}{\frac{1}{2}} / \ \frac{\frac{1}{2}}{\frac{1}{2}} \ \frac{\frac{1}{2}}{\frac{1}{2}} \lgroup \frac{\frac{1}{2}}{\frac{1}{2}} \rgroup \frac{\frac{1}{2}}{\frac{1}{2}} \frac{\frac{1}{2}}{\frac{1}{2}} A B_{T}^{n i 1} C \frac{1}{\sqrt{2} + \frac{1}{\sqrt{3} + \frac{1}{\sqrt{4} + \frac{1}{\sqrt{5} + \frac{1}{\sqrt{6} +}}}}} \frac{1}{\sqrt{2} + \frac{1}{\sqrt{3} + \frac{1}{\sqrt{4} + \frac{1}{\sqrt{5} + \frac{1}{\sqrt{6} +}}}}} (\binom{\sin}{M} (\binom{\sin}{M} (\binom{\sin}{M} (\frac{\sin}{M} (\frac{\sin}{M} (\frac{\sin}{M} (\frac{\sin}{M} (\frac{\sin}{M} (\frac{\sin}{M} \binom{\sin}{M} \binom{\sin}{M} \binom{\sin}{M} \frac{\sin}{M} \frac{\sin}{M} \frac{\sin}{M} \frac{\sin}{M} \frac{\sin}{M} \frac{\sin}{M} \end{matrix}