Elementary Functions - 4.45 Methods of Computation

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
4.45.E1 y = x 2 - m - 1 𝑦 superscript π‘₯ superscript 2 π‘š 1 {\displaystyle{\displaystyle y=x^{2^{-m}}-1}}
y = x^{2^{-m}}-1		 

y = (x)^((2)^(- m))- 1
 
y == (x)^((2)^(- m))- 1
 Skipped - no semantic math Skipped - no semantic math - -
4.45.E2 ln ⁑ x = 2 m ⁒ ln ⁑ ( 1 + y ) π‘₯ superscript 2 π‘š 1 𝑦 {\displaystyle{\displaystyle\ln x=2^{m}\ln\left(1+y\right)}}
\ln@@{x} = 2^{m}\ln@{1+y}		 

ln(x) = (2)^(m)* ln(1 + y)

Log[x] == (2)^(m)* Log[1 + y]
Failure Failure
Failed [54 / 54]
Result: 1.791759469-6.283185308*I
Test Values: {x = 1.5, y = -1.5, m = 1}


Result: 3.178053830-12.56637062*I
Test Values: {x = 1.5, y = -1.5, m = 2}


Result: 5.950642553-25.13274123*I
Test Values: {x = 1.5, y = -1.5, m = 3}


Result: -1.427116356
Test Values: {x = 1.5, y = 1.5, m = 1}

... skip entries to safe data
Failed [54 / 54]
Result: Complex[1.791759469228055, -6.283185307179586]
Test Values: {Rule[m, 1], Rule[x, 1.5], Rule[y, -1.5]}


Result: Complex[3.1780538303479453, -12.566370614359172]
Test Values: {Rule[m, 2], Rule[x, 1.5], Rule[y, -1.5]}

... skip entries to safe data
4.45.E3 ln ⁑ x = ln ⁑ ΞΎ + m ⁒ ln ⁑ 10 π‘₯ πœ‰ π‘š 10 {\displaystyle{\displaystyle\ln x=\ln\xi+m\ln 10}}
\ln@@{x} = \ln@@{\xi}+m\ln@@{10}		 

ln(x) = ln(xi)+ m*ln(10)

Log[x] == Log[\[Xi]]+ m*Log[10]
Failure Failure
Failed [90 / 90]
Result: -1.897119985-.5235987755*I
Test Values: {x = 1.5, xi = 1/2*3^(1/2)+1/2*I, m = 1}


Result: -4.199705078-.5235987755*I
Test Values: {x = 1.5, xi = 1/2*3^(1/2)+1/2*I, m = 2}


Result: -6.502290171-.5235987755*I
Test Values: {x = 1.5, xi = 1/2*3^(1/2)+1/2*I, m = 3}


Result: -1.897119985-2.094395102*I
Test Values: {x = 1.5, xi = -1/2+1/2*I*3^(1/2), m = 1}

... skip entries to safe data
Failed [90 / 90]
Result: Complex[-1.8971199848858815, -0.5235987755982988]
Test Values: {Rule[m, 1], Rule[x, 1.5], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-4.199705077879927, -0.5235987755982988]
Test Values: {Rule[m, 2], Rule[x, 1.5], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
4.45#Ex1 m = ⌊ x ln ⁑ 10 + 1 2 βŒ‹ π‘š π‘₯ 10 1 2 {\displaystyle{\displaystyle m=\left\lfloor\frac{x}{\ln 10}+\frac{1}{2}\right% \rfloor}}
m = \floor{\frac{x}{\ln@@{10}}+\frac{1}{2}}		 

m = floor((x)/(ln(10))+(1)/(2))

m == Floor[Divide[x,Log[10]]+Divide[1,2]]
Failure Failure
Failed [7 / 9]
Result: 1.
Test Values: {x = 1.5, m = 2}


Result: 2.
Test Values: {x = 1.5, m = 3}


Result: 1.
Test Values: {x = .5, m = 1}


Result: 2.
Test Values: {x = .5, m = 2}

... skip entries to safe data
Failed [7 / 9]
Result: 1.0
Test Values: {Rule[m, 2], Rule[x, 1.5]}


Result: 2.0
Test Values: {Rule[m, 3], Rule[x, 1.5]}

... skip entries to safe data
4.45#Ex2 y = x - m ⁒ ln ⁑ 10 𝑦 π‘₯ π‘š 10 {\displaystyle{\displaystyle y=x-m\ln 10}}
y = x-m\ln@@{10}		 

y = x - m*ln(10)

y == x - m*Log[10]
Failure Failure
Failed [54 / 54]
Result: -.697414907
Test Values: {x = 1.5, y = -1.5, m = 1}


Result: 1.605170186
Test Values: {x = 1.5, y = -1.5, m = 2}


Result: 3.907755279
Test Values: {x = 1.5, y = -1.5, m = 3}


Result: 2.302585093
Test Values: {x = 1.5, y = 1.5, m = 1}

... skip entries to safe data
Failed [54 / 54]
Result: -0.6974149070059541
Test Values: {Rule[m, 1], Rule[x, 1.5], Rule[y, -1.5]}


Result: 1.6051701859880918
Test Values: {Rule[m, 2], Rule[x, 1.5], Rule[y, -1.5]}

... skip entries to safe data
4.45.E5 e x = 10 m ⁒ e y superscript 𝑒 π‘₯ superscript 10 π‘š superscript 𝑒 𝑦 {\displaystyle{\displaystyle e^{x}=10^{m}e^{y}}}
e^{x} = 10^{m}e^{y}		 

exp(x) = (10)^(m)* exp(y)
 
Exp[x] == (10)^(m)* Exp[y]
 Skipped - no semantic math Skipped - no semantic math - -
4.45#Ex3 m = ⌊ ΞΎ + 1 2 βŒ‹ π‘š πœ‰ 1 2 {\displaystyle{\displaystyle m=\left\lfloor\xi+\tfrac{1}{2}\right\rfloor}}
m = \floor{\xi+\tfrac{1}{2}}		 

m = floor(xi +(1)/(2))

m == Floor[\[Xi]+Divide[1,2]]
Failure Failure
Failed [26 / 30]
Result: 1.
Test Values: {xi = 1/2*3^(1/2)+1/2*I, m = 2}


Result: 2.
Test Values: {xi = 1/2*3^(1/2)+1/2*I, m = 3}


Result: 1.
Test Values: {xi = -1/2+1/2*I*3^(1/2), m = 1}


Result: 2.
Test Values: {xi = -1/2+1/2*I*3^(1/2), m = 2}

... skip entries to safe data
Failed [26 / 30]
Result: 1.0
Test Values: {Rule[m, 2], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: 2.0
Test Values: {Rule[m, 3], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
4.45#Ex4 ΞΈ = Ο€ ⁒ ( ΞΎ - m ) πœƒ πœ‹ πœ‰ π‘š {\displaystyle{\displaystyle\theta=\pi(\xi-m)}}
\theta = \pi(\xi-m)		 

theta = Pi*(xi - m)
 
\[Theta] == Pi*(\[Xi]- m)
 Skipped - no semantic math Skipped - no semantic math - -
4.45#Ex5 sin ⁑ x = ( - 1 ) m ⁒ sin ⁑ ΞΈ π‘₯ superscript 1 π‘š πœƒ {\displaystyle{\displaystyle\sin x=(-1)^{m}\sin\theta}}
\sin@@{x} = (-1)^{m}\sin@@{\theta}		 

sin(x) = (- 1)^(m)* sin(theta)

Sin[x] == (- 1)^(m)* Sin[\[Theta]]
Failure Failure
Failed [81 / 90]
Result: 1.856475321+.3375964631*I
Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = 1.5, m = 1}


Result: .1385146521-.3375964631*I
Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = 1.5, m = 2}


Result: 1.856475321+.3375964631*I
Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = 1.5, m = 3}


Result: 1.338405873+.3375964631*I
Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = .5, m = 1}

... skip entries to safe data
Failed [81 / 90]
Result: Complex[1.8564753209041922, 0.33759646322287]
Test Values: {Rule[m, 1], Rule[x, 1.5], Rule[ΞΈ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.13851465230391657, -0.33759646322287]
Test Values: {Rule[m, 2], Rule[x, 1.5], Rule[ΞΈ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
4.45#Ex6 cos ⁑ x = ( - 1 ) m ⁒ cos ⁑ ΞΈ π‘₯ superscript 1 π‘š πœƒ {\displaystyle{\displaystyle\cos x=(-1)^{m}\cos\theta}}
\cos@@{x} = (-1)^{m}\cos@@{\theta}		 

cos(x) = (- 1)^(m)* cos(theta)

Cos[x] == (- 1)^(m)* Cos[\[Theta]]
Failure Failure
Failed [84 / 90]
Result: .8012802206-.3969495503*I
Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = 1.5, m = 1}


Result: -.6598058172+.3969495503*I
Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = 1.5, m = 2}


Result: .8012802206-.3969495503*I
Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = 1.5, m = 3}


Result: 1.608125581-.3969495503*I
Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = .5, m = 1}

... skip entries to safe data
Failed [84 / 90]
Result: Complex[0.8012802207249281, -0.3969495502290325]
Test Values: {Rule[m, 1], Rule[x, 1.5], Rule[ΞΈ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.6598058173895223, 0.3969495502290325]
Test Values: {Rule[m, 2], Rule[x, 1.5], Rule[ΞΈ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
4.45.E8 2 ⁒ arctan ⁑ x 1 + ( 1 + x 2 ) 1 / 2 = arctan ⁑ x 2 π‘₯ 1 superscript 1 superscript π‘₯ 2 1 2 π‘₯ {\displaystyle{\displaystyle 2\operatorname{arctan}\frac{x}{1+(1+x^{2})^{1/2}}% =\operatorname{arctan}x}}
2\atan@@{\frac{x}{1+(1+x^{2})^{1/2}}} = \atan@@{x}		 0 < x , x < ∞ formulae-sequence 0 π‘₯ π‘₯ {\displaystyle{\displaystyle 0<x,x<\infty}} 
2*arctan((x)/(1 +(1 + (x)^(2))^(1/2))) = arctan(x)

2*ArcTan[Divide[x,1 +(1 + (x)^(2))^(1/2)]] == ArcTan[x]
Successful Failure - Successful [Tested: 3]
4.45.E9 x n = x n - 1 1 + ( 1 + x n - 1 2 ) 1 / 2 subscript π‘₯ 𝑛 subscript π‘₯ 𝑛 1 1 superscript 1 subscript superscript π‘₯ 2 𝑛 1 1 2 {\displaystyle{\displaystyle x_{n}=\frac{x_{n-1}}{1+(1+x^{2}_{n-1})^{1/2}}}}
x_{n} = \frac{x_{n-1}}{1+(1+x^{2}_{n-1})^{1/2}}		 

x[n] = (x[n - 1])/(1 +(1 + (x[n - 1])^(2))^(1/2))
 
Subscript[x, n] == Divide[Subscript[x, n - 1],1 +(1 + (Subscript[x, n - 1])^(2))^(1/2)]
 Skipped - no semantic math Skipped - no semantic math - -
4.45.E10 arctan ⁑ x = 2 n ⁒ arctan ⁑ x n π‘₯ superscript 2 𝑛 subscript π‘₯ 𝑛 {\displaystyle{\displaystyle\operatorname{arctan}x=2^{n}\operatorname{arctan}x% _{n}}}
\atan@@{x} = 2^{n}\atan@@{x_{n}}		 

arctan(x) = (2)^(n)* arctan(x[n])

ArcTan[x] == (2)^(n)* ArcTan[Subscript[x, n]]
Failure Failure
Failed [90 / 90]
Result: -.5880026038-.5493061442*I
Test Values: {x = 1.5, x[n] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: -2.158798931-1.098612288*I
Test Values: {x = 1.5, x[n] = 1/2*3^(1/2)+1/2*I, n = 2}


Result: -5.300391585-2.197224577*I
Test Values: {x = 1.5, x[n] = 1/2*3^(1/2)+1/2*I, n = 3}


Result: 2.553590050-1.316957897*I
Test Values: {x = 1.5, x[n] = -1/2+1/2*I*3^(1/2), n = 1}

... skip entries to safe data
Failed [90 / 90]
Result: Complex[-0.5880026035475677, -0.5493061443340551]
Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[Subscript[x, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.1587989303424644, -1.0986122886681102]
Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[Subscript[x, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
4.45#Ex7 x 1 = 0.90000 ⁒ … subscript π‘₯ 1 0.90000 … {\displaystyle{\displaystyle x_{1}=0.90000\dots}}
x_{1} = 0.90000\dots		 

x[1] = 0.90000
 
Subscript[x, 1] == 0.90000
 Skipped - no semantic math Skipped - no semantic math - -
4.45#Ex8 x 2 = 0.38373 ⁒ … subscript π‘₯ 2 0.38373 … {\displaystyle{\displaystyle x_{2}=0.38373\dots}}
x_{2} = 0.38373\dots		 

x[2] = 0.38373
 
Subscript[x, 2] == 0.38373
 Skipped - no semantic math Skipped - no semantic math - -
4.45#Ex9 x 3 = 0.18528 ⁒ … subscript π‘₯ 3 0.18528 … {\displaystyle{\displaystyle x_{3}=0.18528\dots}}
x_{3} = 0.18528\dots		 

x[3] = 0.18528
 
Subscript[x, 3] == 0.18528
 Skipped - no semantic math Skipped - no semantic math - -
4.45#Ex10 x 4 = 0.09185 ⁒ … subscript π‘₯ 4 0.09185 … {\displaystyle{\displaystyle x_{4}=0.09185\dots}}
x_{4} = 0.09185\dots		 

x[4] = 0.09185
 
Subscript[x, 4] == 0.09185
 Skipped - no semantic math Skipped - no semantic math - -
4.45.E13 arctan ⁑ x = 16 ⁒ arctan ⁑ x 4 π‘₯ 16 subscript π‘₯ 4 {\displaystyle{\displaystyle\operatorname{arctan}x=16\operatorname{arctan}x_{4% }}}
\atan@@{x} = 16\atan@@{x_{4}}		 

arctan(x) = 16*arctan(x[4])

ArcTan[x] == 16*ArcTan[Subscript[x, 4]]
Failure Failure
Failed [30 / 30]
Result: -11.58357690-4.394449154*I
Test Values: {x = 1.5, x[4] = 1/2*3^(1/2)+1/2*I}


Result: 13.54916434-10.53566318*I
Test Values: {x = 1.5, x[4] = -1/2+1/2*I*3^(1/2)}


Result: -11.58357690+10.53566318*I
Test Values: {x = 1.5, x[4] = 1/2-1/2*I*3^(1/2)}


Result: 13.54916434+4.394449154*I
Test Values: {x = 1.5, x[4] = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[-11.583576891111845, -4.394449154672441]
Test Values: {Rule[x, 1.5], Rule[Subscript[x, 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[13.549164337606502, -10.535663175398536]
Test Values: {Rule[x, 1.5], Rule[Subscript[x, 4], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.45.E13 16 ⁒ arctan ⁑ x 4 = 1.46563 ⁒ … 16 subscript π‘₯ 4 1.46563 … {\displaystyle{\displaystyle 16\operatorname{arctan}x_{4}=1.46563\dots}}
16\atan@@{x_{4}} = 1.46563\dots		 

16*arctan(x[4]) = 1.46563

16*ArcTan[Subscript[x, 4]] == 1.46563
Failure Failure
Failed [10 / 10]
Result: 11.10074062+4.394449154*I
Test Values: {x[4] = 1/2*3^(1/2)+1/2*I}


Result: -14.03200062+10.53566318*I
Test Values: {x[4] = -1/2+1/2*I*3^(1/2)}


Result: 11.10074062-10.53566318*I
Test Values: {x[4] = 1/2-1/2*I*3^(1/2)}


Result: -14.03200062-4.394449154*I
Test Values: {x[4] = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [10 / 10]
Result: Complex[11.100740614359175, 4.394449154672441]
Test Values: {Rule[Subscript[x, 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-14.032000614359173, 10.535663175398536]
Test Values: {Rule[Subscript[x, 4], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.45.E15 ln ⁑ z = ln ⁑ | z | + i ⁒ ph ⁑ z 𝑧 𝑧 𝑖 phase 𝑧 {\displaystyle{\displaystyle\ln z=\ln|z|+i\operatorname{ph}z}}
\ln@@{z} = \ln@@{|z|}+i\phase@@{z}		 - Ο€ ≀ ph ⁑ z , ph ⁑ z ≀ Ο€ formulae-sequence πœ‹ phase 𝑧 phase 𝑧 πœ‹ {\displaystyle{\displaystyle-\pi\leq\operatorname{ph}z,\operatorname{ph}z\leq% \pi}} 
ln(z) = ln(abs(z))+ I*argument(z)
 
Log[z] == Log[Abs[z]]+ I*Arg[z]
 Failure Successful Successful [Tested: 7] Successful [Tested: 7]
4.45.E16 e z = e β„œ ⁑ z ⁒ ( cos ⁑ ( β„‘ ⁑ z ) + i ⁒ sin ⁑ ( β„‘ ⁑ z ) ) superscript 𝑒 𝑧 superscript 𝑒 𝑧 𝑧 𝑖 𝑧 {\displaystyle{\displaystyle e^{z}=e^{\Re z}(\cos\left(\Im z\right)+i\sin\left% (\Im z\right))}}
e^{z} = e^{\realpart@@{z}}(\cos@{\imagpart@@{z}}+i\sin@{\imagpart@@{z}})		 

exp(z) = exp(Re(z))*(cos(Im(z))+ I*sin(Im(z)))
 
Exp[z] == Exp[Re[z]]*(Cos[Im[z]]+ I*Sin[Im[z]])
 Failure Successful Successful [Tested: 7] Successful [Tested: 7]
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