MathML TestSuite, Version 2: complex3

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File:	/TortureTests/Complexity/complex3.xml
Author:	Mackichan (S. Swanson)
Description:	around 300 equation tests
Sample Rendering:	N/A

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M1

2 \sum a b

M2

x^{' 3}

M3

f^{'} (x) + \sin \cos θ = 1

M4

f (z) = \sum_{n = 0}^{\infty} a_{n} z^{n}, | z | < R ​ (R \neq 0)

M5

\int_{C} (\sum_{n = 0}^{\infty} a_{n} z^{n}) ⅆ z = \sum_{n = 0}^{\infty} a_{n} \int_{C} z^{n} ⅆ z

M6

lim_{n \to \infty} | \int_{C} [f (z) - \sum_{k = 0}^{n} a_{k} z^{k}] ⅆ z | = 0

M7

n \geq N (ε) \Rightarrow | f (z) - \sum_{k = 0}^{n} a_{k} z^{k} | < ε

M8

10 Bq + 10 Ci

M9

10 amol + 10 Emol - 10 fmol + 10 Gmol - 10 kmol + 10 Mmol

M10

10 μmol + 10 mmol - 10 mol + 10 nmol - 10 Pmol + 10 pmol - 10 Tmol

M11

10 acre + 10 hectare - 10 {ft}^{2} + 10 {in}^{2} - 10 m^{2}

M12

10 A + 10 kA - 10 μA + 10 mA - 10 nA

M13

10 F + 10 μF - 10 mF + 10 nF - 10 pF

M14

10 C + 1.0 m/s/s - 0.1 m / s^{2}

M15

10 kS + 10 μS - 10 mS + 10 S

M16

10 kV + 10 MV - 10 μV + 10 mV - 10 nV + 10 pV - 10 V

M17

10 GΩ + 10 kΩ - 10 MΩ + 10 mΩ - 10 Ω

M18

10 Btu + 10 cal - 10 eV + 10 erg - 10 GeV + 10 GJ

M19

10 J + 10 kcal - 10 kJ + 10 MeV - 10 MJ + 10 μJ - 10 mJ + 10 nJ

M20

10 dyn + 10 kN - 10 MN + 10 μN - 10 mN + 10 N - 10 ozf + 10 lbf

M21

10 EHz + 10 GHz - 10 Hz + 10 kHz - 10 MHz + 10 PHz - 10 THz

M22

10 fc + 10 lx - 10 phot

M23

10 Å + 10 am - 10 cm + 10 dm - 10 fm + 10 ft - 10 in

M24

10 km + 10 m - 10 μm + 10 mi - 10 mm + 10 nm - 10 pm

M25

10 sb

M26

10 lm

M27

10 cd

M28

10 Mx + 10 μWb - 10 mWb + 10 nWb - 10 Wb

M29

10 G + 10 μT - 10 mT + 10 nT - 10 pT + 10 T

M30

10 H + 10 μH - 10 mH

M31

10 u + 10 cg - 10 dg + 10 g - 10 kg + 10 μg - 10 mg + 10 lb - 10 slug

M32

10 ° + 10 μrad - 10 mrad + 10 ​^{′} - 10 rad + 10 ​^{′′}

M33

10 GW + 10 hp - 10 kW + 10 MW - 10 μW + 10 mW - 10 nW + 10 W

M34

10 atm + 10 bar - 10 kbar + 10 kPa - 10 MPa + 10 μPa - 10 mbar + 10 mmHg - 10 Pa + 10 torr

M35

10 sr

M36

10 °C + 10 °F - 10 K

M37

10 as + 10 d - 10 fs + 10 h - 10 μs + 10 ms - 10 min + 10 ns - 10 ps + 10 s - 10 y

M38

10 {ft}^{3} + 10 {in}^{3} - 10 m^{3} + 10 gal - 10 l

M39

10 ml + 10 pint - 10 qt

M40

\frac{1}{x (y)} = (- \int e^{- \frac{1}{2} y^{2}} \sin y ⅆ y + C_{1}) e^{\frac{1}{2} y^{2}}

M41

ⅅ_{x} y - y = \sin x

M42

(\binom{1}{2}) (\binom{1}{2}) (\binom{1}{2})

M43

[\binom{1}{2}] (\binom{1}{2}) {\binom{1}{2}}

M44

〈 \frac{1}{2} 〉 ⌊ \frac{1}{2} ⌋ ⌈ \frac{1}{2} ⌉

M45

↑ \frac{1}{2} ↑ ↓ \frac{1}{2} ↓ ↕ \frac{1}{2} ↕

M46

\frac{1}{2} \frac{1}{2} \frac{1}{2}

M47

\frac{1}{2} \frac{1}{2} \frac{1}{2}

M48

\binom{1}{2} \binom{1}{2} \binom{1}{2}

M49

- (a - b) = b - a

M50

\frac{2}{5} + \frac{3}{7} = \frac{2 \cdot 7 + 3 \cdot 5}{35} = \frac{29}{35}

M51

| a | = {\begin{aligned} a & if & a \geq 0 \\ - a & if & a < 0 \end{aligned}

M52

a^{n} = \underset{n factors}{\underset{︸}{a \cdot a \cdot \dots \cdot a}}

M53

{(\frac{a}{b})}^{- n} = {(\frac{b}{a})}^{n}

M54

\sqrt[n]{a} = b means b^{n} = a .

M55

\sqrt[4]{\frac{16}{81}} = \frac{\sqrt[4]{16}}{\sqrt[4]{81}} = \frac{2}{3}

M56

{x ∣ x \neq 0, x \neq 1}

M57

a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}

M58

a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})

M59

{(x + y)}^{2}

M60

H = {(\begin{matrix} a & b \\ c & d \end{matrix}) \in G ∣ a d - b c = 1}

M61

| x | + || y || + {z} - [a c] + (b) = [a, b]

M62

x = 1

M63

x = 1

M64

x = 1

M65

x = 1

M66

[- \frac{10}{3}, - \frac{7}{3}) \cup (- \frac{7}{3}, - \frac{4}{3}]

M67

A \frac{\partial u}{\partial x} + B \frac{\partial u}{\partial y} + C u = E

M68

\sum_{​}^{​} x

M69

\sum_{\begin{matrix} 1 < i < 10 \\ 1 < j < 10 \end{matrix}}^{​} 2^{i + j}

M70

Γ_{1_{^{\begin{matrix} 2_{^{\begin{matrix} 3 \\ 4 \end{matrix}}}^{​} \\ 5_{^{\begin{matrix} 6 \\ 7 \end{matrix}}}^{​} \end{matrix}}}^{​}}^{1^{\begin{matrix} 5^{\begin{matrix} 7 \\ 6 \end{matrix}} \\ 2^{\begin{matrix} 4 \\ 3 \end{matrix}} \end{matrix}}}

M71

y (x) = \frac{x e^{x} - e^{x} + 2}{e^{x}} = x - 1 + \frac{2}{e^{x}}

M72

\begin{array}{r} ⅅ_{x ​ x} y - y = 0 \\ y (0) = 1 \\ y^{'} (0) = 0 \end{array}

M73

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

M74

y (t) = 2 \tan (2 t - \frac{1}{4} π)

M75

ℱ (\begin{matrix} e^{2 π i x} \\ 2 π Dirac (x - 2 π) \end{matrix}, x, s) = (\begin{matrix} 2 π Dirac (s - 2 π) \\ 2 π e^{- 2 i π s} \end{matrix})

M76

\begin{matrix} x = 1 \\ x + 3 = 123 \end{matrix}

M77

\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}

M78

K_{v} (z) = {BesselK}_{v} (z)

M79

z^{2} \frac{ⅆ^{2} w}{ⅆ z^{2}} + z \frac{ⅆ w}{ⅆ z} - (z^{2} + v^{2}) w = 0

M80

\frac{\partial^{2} u (x, y)}{\partial x^{2}} - \frac{\partial^{2} u (x, y)}{\partial y^{2}} = 0

M81

y (t, x) = F_{1} (- x - a t) + F_{2} (x - a t)

M82

\begin{array}{lcr} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}

M83

2 x + 1 = 5

M84

\begin{matrix} 1 = 3 \\ 9 = 7 \end{matrix}

M85

\begin{matrix} a b \\ c d \\ e f \end{matrix}

M86

\begin{matrix} x + 2 y - 3 = 5 \\ 4 x - y - 5 = 98 \end{matrix}

M87

\begin{matrix} x = z \\ 1 = 3 \end{matrix}

M88

\begin{matrix} A_{1} = N_{0} (λ; Ω^{'}) - φ (λ; Ω^{'}), \\ A_{2} = φ (λ; Ω^{'}) - φ (λ; Ω), \\ A_{3} = N (λ; ω) . \end{matrix}

M89

\begin{matrix} \sin θ \\ \cos γ \end{matrix}

M90

x = {\begin{cases} x & if x < 0 \\ - x & if x \geq 0 \end{cases}

M91

\begin{matrix} L M R M \\ L M R M \end{matrix}

M92

\begin{matrix} M A T H \\ M A T H \end{matrix}

M93

M94

⋮

M95

\nabla\times F = 0

M96

\nabla\cdot F

M97

\nabla\cdot\nabla F = \nabla^{2} F + 7 = A

M98

\nabla\times (x y, y z, z x) = [\begin{matrix} - y \\ - z \\ - x \end{matrix}]

M99

\nabla\times (y, z, x) = (- 1, - 1, - 1) \neq 0

M100

x + y + α = 102

M101

a + b = c

M102

x + 1

M103

x + f (x) - 1 = 123

M104

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 .

M105

(\frac{\partial f}{\partial x_{1}} (c_{1}, c_{2}, \dots, c_{n}), \frac{\partial f}{\partial x_{2}} (c_{1}, c_{2}, \dots, c_{n}), \dots, \frac{\partial f}{\partial x_{n ​ 1}} (c_{1}, c_{2}, \dots, c_{n}))

M106

\nabla (c u v + v^{2} w) = (u v, c v, c u + 2 v w, v^{2})

M107

\begin{matrix} D_{u} f (a, b, c) = \nabla f (a, b, c) \cdot u \\ = \frac{\partial f}{\partial x} (a, b, c) u_{1} + \frac{\partial f}{\partial y} (a, b, c) u_{2} + \frac{\partial f}{\partial z} (a, b, c) u_{3} \end{matrix}

M108

θ \in {π + 2 X_{3} π - (\arccos \frac{1}{7} \sqrt{14}) | X_{3} \in ℤ}, θ \in {2 X_{4} π - π + (\arccos \frac{1}{7} \sqrt{14}) | X_{4} \in ℤ}

M109

P = A {(A^{T} A)}^{- 1} A^{T}

M110

det (\begin{matrix} x & y & 1 \\ a & b & 1 \\ a & d & 1 \end{matrix}) = x b - x d + a d - a b = 0

M111

A (θ) A (- θ) = [\begin{array}{rr} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{array}] [\begin{array}{rr} \cos θ & \sin θ \\ - \sin θ & \cos θ \end{array}]

M112

J (A) = [\begin{matrix} J_{n_{1}} (λ_{1}) & 0 & \dots & 0 \\ 0 & J_{n_{2}} (λ_{2}) & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & J_{n_{k}} (λ_{k}) \end{matrix}]

M113

det (\begin{matrix} - 4 + X & - 1 & 0 \\ 0 & - 4 + X & 0 \\ 0 & 0 & - 4 + X \end{matrix}) = {(X - 4)}^{3}

M114

{(\begin{matrix} - \frac{1}{2} - \frac{1}{6} \sqrt{33} \\ 1 \end{matrix})} \leftrightarrow \frac{5}{2} - \frac{1}{2} \sqrt{33}

M115

∥ A ∥ = max_{x \neq 0} \frac{∥ A x ∥}{∥ x ∥}

M116

(\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})

M117

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}]

M118

x = lim_{x = 1} \sum_{1}^{2} a

M119

\int_{a}^{b} f (x) ⅆ x = lim_{∥ P ∥ \to 0} \sum_{i = 1}^{n} f ({\bar{x}}_{i}) Δ x_{i}

M120

\int_{a}^{b} f (x) ⅆ x = lim_{n \to \infty} \frac{b - a}{n} \sum_{i = 1}^{n} f (a + i \frac{b - a}{n})

M121

\int_{0}^{2} x^{5} \sqrt{x^{3} + 1} ⅆ x = \int_{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}}) ⅆ u

M122

\int f (g (x)) g^{'} (x) ⅆ x = \int f (u) ⅆ u

M123

x = 2 \sum_{n = 1}^{100} n (n - 1)

M124

lim_{x \to 0} \sin (\frac{1}{x}) = - 1 .. 1

M125

h (i, j) = (2 - j) g (i) + (j - 1) f (g (i))

M126

△ : [0, 1] \to [0, 1]

M127

0 ▽ x = x

M128

x △ y = h^{- 1} (h (x) h (y))

M129

x △ y = f^{- 1} (max {f (x) + f (y) - 1, 0})

M130

x ▽ y = η (η (x) △ η (y))

M131

x △_{0} y = {\begin{matrix} x \land y & if & x \lor y = 1 \\ 0 & if & x \lor y < 1 \end{matrix}

M132

lim_{a \to 1^{+}} \log_{a} [1 + \frac{(a^{x} - 1) (a^{y} - 1)}{a - 1}] = lim_{a \to 1^{-}} \log_{a} [1 + \frac{(a^{x} - 1) (a^{y} - 1)}{a - 1}] = x y

M133

g (x) = \exp (- \frac{1 - {(1 - x)}^{a}}{(2^{a} - 1) {(1 - x)}^{a}})

M134

Aut (I) = {f : [0, 1] \to [0, 1] | \begin{array}{l} f is one-to-one and onto, and \\ x \leq y implies f (x) \leq f (y) \end{array}}

M135

x^{2} + y^{2} = r^{2}, \tan θ = \frac{y}{x}

M136

\sqrt{2} \sqrt{1 - t^{2}}

M137

[(2 + \sin t) 10 \cos t, (2 + \cos t) 10 \sin t, 3 \sin 3 t]

M138

{t = 0, s = 0}, {t = π, s = π}

M139

\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}

M140

\begin{matrix} testing x^{2} end. \end{matrix}

M141

\begin{matrix} x \end{matrix}

M142

\begin{matrix} x \end{matrix}

M143

\begin{matrix} x \end{matrix}

M144

\begin{matrix} x \end{matrix}

M145

\frac{ⅆ f}{ⅆ x} (x_{1}) = 5

M146

\int x ⅆ x = \iint x y ⅆ x ⅆ y = ∭ x y z ⅆ x ⅆ y ⅆ z = ⨌ x y z t ⅆ x ⅆ y ⅆ z ⅆ t

M147

mod a

M148

5 mod 3 = 2

M149

f (0) mod 3 = 1

M150

5 x + 4 \equiv 8 ​ (mod 13)

M151

a = (5 - 3) / 5 mod 7 = 6

M152

(2 x^{2} + x + 2) + (2 x + 1) mod 3 = 2 x^{2}

M153

\begin{array}{rcl} + & 0 & 1 \\ 000 & 000 & 111 \\ 1 & 1 & 0 \end{array}

M154

4. 974 9

M155

\frac{ⅆ}{ⅆ x} F (x)

M156

[86.333, 146.33, 129.33]

M157

BinomialDist (x; n, p) = \sum_{k = 0}^{x} (\binom{n}{k}) p^{k} q^{n - k}

M158

Pr (X \leq 54) = BinomialDist (54; 100, .55) = .45846

M159

k = max {| \frac{\partial f}{\partial y} (x, y) | : (x, y) \in D} .

M160

m = lim_{x \overset{}{\to} a} \frac{f (x) - f (a)}{x - a}

M161

| A | = | \begin{array}{llllll} a_{1 ​ 1} & a_{1 ​ 2} & \cdot & \cdot & \cdot & a_{1 ​ n} \\ a_{2 ​ 1} & a_{2 ​ 2} & \cdot & \cdot & \cdot & a_{2 ​ n} \\ \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot \\ a_{n ​ 1} & a_{n ​ 2} & \cdot & \cdot & \cdot & a_{n ​ n} \end{array} | = a_{1 ​ 1} A_{1 ​ 1} + a_{1 ​ 2} A_{1 ​ 2} + \dots + a_{1 ​ n} A_{1 ​ n}

M162

x = 1 (hl text x end.)

M163

x = 1 (hl to URI x end)

M164

x = 1 (sex)

M165

x = 1 (jbm)

M167

f (x) g [y] h {z} + ⌊ a ⌋ ⌈ b ⌉ 〈 c 〉

M168

\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}} ⇓

M169

x \begin{array}{ll} x & x \\ x & x \end{array} x

M170

(a_{1}, a_{2}, \dots, a_{n}) \cdot (b_{1}, b_{2}, \dots, b_{n}) = a_{1} b_{1}^{*} + a_{2} b_{2}^{*} + \dots + a_{n} b_{n}^{*}

M171

⌊ \frac{n}{5} ⌋ + ⌊ \frac{n}{5^{2}} ⌋ + ⌊ \frac{n}{5^{3}} ⌋ + ⌊ \frac{n}{5^{4}} ⌋ + \dots

M172

x_{1} + \dots + x_{n}

M173

\underset{k times}{\underset{︸}{x + \dots + x}}

M174

\sqrt[n]{x_{1} x_{2} \dots x_{n}}

M175

n! = 1 \times 2 \times 3 \times 4 \times \dots \times n

M176

P : a = x_{0} < x_{1} < x_{2} < \dots < x_{n} = b

M177

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} π))}

M178

\int \cos (A x) \sin (B x) ⅆ x = \frac{- \cos (B - A) x}{2 (B - A)} + \frac{- \cos (B + A) x}{2 (B + A)} + C .

M179

235.3 + 813 = 1048. 3

M180

max_{- 2 \leq x \leq 2} (x^{3} - 6 x + 3) = 8.0

M181

x decade = 2 century

M182

\frac{ⅆ^{5} (x^{7} - 3 x^{6})}{ⅆ x^{5}} \frac{ⅆ^{n} \sin x}{ⅆ x^{n}} \frac{ⅆ^{3}}{ⅆ x^{3}} f (x) \frac{ⅆ^{2}}{ⅆ t^{2}} (4 t^{5} - 3 t)

M183

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} π))}

M184

\int_{R^{3}} (\frac{{| u_{1} |}^{2} + {| \nabla u_{0} |}^{2}}{2} + \frac{{| u_{0} |}^{6}}{6}) ⅆ x < \infty

M185

(\nabla\times F) \cdot k = z + 1

M186

M \frac{M^{\frac{M}{M}}}{M}

M187

ⅅ_{x} x^{2} ⅅ_{x} (x^{2}) ⅅ_{x ​ x} (x^{2}) ⅅ_{x^{2}} (x^{2}) ⅅ_{x ​ y} (x^{2} y^{3}) ⅅ_{x^{s} ​ y^{t}} (x^{2} y^{3})

M188

5 24! x^{6}

M189

\begin{matrix} \begin{matrix} x + \sqrt[2]{\frac{a^{y - 1}}{12.34}} & \sin θ \\ 1 \end{matrix} \end{matrix}

M190

\begin{array}{c} 0 & 1 \\ 1 & 0 \end{array}

M191

(\begin{array}{c} 0 & - i \\ i & 0 \end{array})

M192

[\begin{array}{c} 1 & 0 \\ 0 & - 1 \end{array}]

M193

| \begin{array}{c} a & b \\ c & d \end{array} |

M194

∥ \begin{array}{c} 1 & 0 & 1 \\ 0 & 11 \end{array} ∥

M195

\begin{array}{c} 1 & 2 & 3 \\ 4 & 5 \end{array}

M196

testing \begin{matrix} \sin θ \end{matrix}

M197

\hat{a} + \overset{ˇ}{b} + \tilde{c} + \overset{´}{d} + \overset{`}{e} + \overset{˘}{f} + \bar{g} + h + \overset{˚}{i} + \dot{j} + \ddot{k} + \overset{⃛}{l} + \overset{⃜}{m} + \vec{n}

M198

f (g (x)) = \sin^{3} x^{2} + \sin x^{2} \sin (\sin x^{2})

M199

(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) \cdot (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) \cdot (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{ⅆ}{ⅆ 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

∠ α + ∠ A B C + ∠ 1 = ▵ a b c

y = e^{- \int P ⅆ x} [\int e^{\int P ⅆ x} Q ⅆ x + c]

x = 1 + y^{3}

and

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 \neq b \neq x

c ≮ d ≮ y

e ≯ f ≯ 11

g \notin h \notin Z

k ≁ l ≁ 3

A ⊄ B \subset C

A ⊈ B ⊈ C

10 ≢ 11 \equiv 12

x ≰⃥ y ≰⃥ z

\lim^{¯} x

\lim_{̲} x

\lim_{\to} x

\lim_{\leftarrow} 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}} ↑ \sum_{1}^{2}}

〈 \frac{\frac{1}{2}}{\frac{1}{2}} | \sum_{1}^{2} 〉

⌈ \frac{\frac{1}{2}}{\frac{1}{2}} | \sum_{1}^{2} ⌉

⇓ \frac{\frac{1}{2}}{\frac{1}{2}} ↕ \sum_{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 \leftarrow_{}^{n + μ - 1} B \to_{T}^{n \pm i - 1} C

\frac{1}{\sqrt{2} + \frac{1}{\sqrt{3} + \frac{1}{\sqrt{4} + \frac{1}{\sqrt{5} + \frac{1}{\sqrt{6} + \dots}}}}}

\frac{1}{\sqrt{2} + \frac{1}{\sqrt{3} + \frac{1}{\sqrt{4} + \frac{1}{\sqrt{5} + \frac{1}{\sqrt{6} + \dots}}}}}

(\binom{\sin θ}{M} ⌋

(\binom{\sin θ}{M} ⌋

(\binom{\sin θ}{M} ⌋

(\frac{\sin θ}{M} ⌋

(\frac{\sin θ}{M} ⌋

(\frac{\sin θ}{M} ⌋

(\frac{\sin θ}{M} ⌋

(\frac{\sin θ}{M} ⌋

(\frac{\sin θ}{M} ⌋

\binom{\sin θ}{M}

\binom{\sin θ}{M}

\binom{\sin θ}{M}

\frac{\sin θ}{M}

\frac{\sin θ}{M}

\frac{\sin θ}{M}

\frac{\sin θ}{M}

\frac{\sin θ}{M}

\frac{\sin θ}{M}

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