Numerical Methods - 4.2 Definitions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
4.2.E1 Ln ⁑ z = ∫ 1 z d t t multivalued-natural-logarithm 𝑧 superscript subscript 1 𝑧 𝑑 𝑑 {\displaystyle{\displaystyle\operatorname{Ln}z=\int_{1}^{z}\frac{\mathrm{d}t}{% t}}}
\Ln@@{z} = \int_{1}^{z}\frac{\diff{t}}{t}		 z β‰  0 𝑧 0 {\displaystyle{\displaystyle z\neq 0}} 
ln(z) = int((1)/(t), t = 1..z)

Log[z] == Integrate[Divide[1,t], {t, 1, z}, GenerateConditions->None]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
4.2.E2 ln ⁑ z = ∫ 1 z d t t 𝑧 superscript subscript 1 𝑧 𝑑 𝑑 {\displaystyle{\displaystyle\ln z=\int_{1}^{z}\frac{\mathrm{d}t}{t}}}
\ln@@{z} = \int_{1}^{z}\frac{\diff{t}}{t}		 

ln(z) = int((1)/(t), t = 1..z)

Log[z] == Integrate[Divide[1,t], {t, 1, z}, GenerateConditions->None]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
4.2.E3 ln ⁑ z = ln ⁑ | z | + i ⁒ ph ⁑ z 𝑧 𝑧 imaginary-unit phase 𝑧 {\displaystyle{\displaystyle\ln z=\ln\left|z\right|+\mathrm{i}\operatorname{ph% }z}}
\ln@@{z} = \ln@@{\abs{z}}+\iunit\phase@@{z}		 - Ο€ < ph ⁑ z , ph ⁑ z < Ο€ formulae-sequence πœ‹ phase 𝑧 phase 𝑧 πœ‹ {\displaystyle{\displaystyle-\pi<\operatorname{ph}z,\operatorname{ph}z<\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]
4.2.E4 z = x 𝑧 π‘₯ {\displaystyle{\displaystyle z=x}}
z = x		 - ∞ < x , x < 0 formulae-sequence π‘₯ π‘₯ 0 {\displaystyle{\displaystyle-\infty<x,x<0}} 
(x + y*I) = x
 
(x + y*I) == x
 Skipped - no semantic math Skipped - no semantic math - -
4.2.E5 ln ⁑ z = ln ⁑ | z | + i ⁒ ph ⁑ z 𝑧 𝑧 imaginary-unit phase 𝑧 {\displaystyle{\displaystyle\ln z=\ln\left|z\right|+\mathrm{i}\operatorname{ph% }z}}
\ln@@{z} = \ln@@{\abs{z}}+\iunit\phase@@{z}		 - Ο€ < ph ⁑ z , ph ⁑ z ≀ Ο€ formulae-sequence πœ‹ phase 𝑧 phase 𝑧 πœ‹ {\displaystyle{\displaystyle-\pi<\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]
4.2.E6 Ln ⁑ z = ln ⁑ z + 2 ⁒ k ⁒ Ο€ ⁒ i multivalued-natural-logarithm 𝑧 𝑧 2 π‘˜ πœ‹ imaginary-unit {\displaystyle{\displaystyle\operatorname{Ln}z=\ln z+2k\pi\mathrm{i}}}
\Ln@@{z} = \ln@@{z}+2k\pi\iunit		 

ln(z) = ln(z)+ 2*k*Pi*I

Log[z] == Log[z]+ 2*k*Pi*I
Failure Failure
Failed [21 / 21]
Result: -6.283185308*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -12.56637062*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 2}


Result: -18.84955592*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 3}


Result: -6.283185308*I
Test Values: {z = -1/2+1/2*I*3^(1/2), k = 1}

... skip entries to safe data
Failed [21 / 21]
Result: Complex[0.0, -6.283185307179586]
Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.0, -12.566370614359172]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
4.2.E7 ln ⁑ ( x + i ⁒ 0 ) = ln ⁑ | x | + i ⁒ Ο€ π‘₯ imaginary-unit 0 π‘₯ 𝑖 πœ‹ {\displaystyle{\displaystyle\ln\left(x+\mathrm{i}0\right)=\ln|x|+i\pi}}
\ln@{x+\iunit 0} = \ln@@{|x|}+ i\pi		 - ∞ < x , x < 0 formulae-sequence π‘₯ π‘₯ 0 {\displaystyle{\displaystyle-\infty<x,x<0}} 
ln(x + I*0) = ln(abs(x))+ I*Pi

Log[x + I*0] == Log[Abs[x]]+ I*Pi
Failure Successful Error Skip - symbolical successful subtest
4.2.E7 ln ⁑ ( x - i ⁒ 0 ) = ln ⁑ | x | - i ⁒ Ο€ π‘₯ imaginary-unit 0 π‘₯ 𝑖 πœ‹ {\displaystyle{\displaystyle\ln\left(x-\mathrm{i}0\right)=\ln|x|-i\pi}}
\ln@{x-\iunit 0} = \ln@@{|x|}- i\pi		 - ∞ < x , x < 0 formulae-sequence π‘₯ π‘₯ 0 {\displaystyle{\displaystyle-\infty<x,x<0}} 
ln(x - I*0) = ln(abs(x))- I*Pi

Log[x - I*0] == Log[Abs[x]]- I*Pi
Failure Failure Error Skip - No test values generated
4.2.E8 log a ⁑ z = ln ⁑ z / ln ⁑ a π‘Ž 𝑧 𝑧 π‘Ž {\displaystyle{\displaystyle\operatorname{log}_{a}z=\ifrac{\ln z}{\ln a}}}
\genlog{a}@@{z} = \ifrac{\ln@@{z}}{\ln@@{a}}		 

log[a](z) = (ln(z))/(ln(a))

Log[a,z] == Divide[Log[z],Log[a]]
Successful Successful - Successful [Tested: 42]
4.2.E9 log a ⁑ z = log b ⁑ z log b ⁑ a π‘Ž 𝑧 𝑏 𝑧 𝑏 π‘Ž {\displaystyle{\displaystyle\operatorname{log}_{a}z=\frac{\operatorname{log}_{% b}z}{\operatorname{log}_{b}a}}}
\genlog{a}@@{z} = \frac{\genlog{b}@@{z}}{\genlog{b}@@{a}}		 

log[a](z) = (log[b](z))/(log[b](a))

Log[a,z] == Divide[Log[b,z],Log[b,a]]
Successful Successful - Successful [Tested: 252]
4.2.E10 log a ⁑ b = 1 log b ⁑ a π‘Ž 𝑏 1 𝑏 π‘Ž {\displaystyle{\displaystyle\operatorname{log}_{a}b=\frac{1}{\operatorname{log% }_{b}a}}}
\genlog{a}@@{b} = \frac{1}{\genlog{b}@@{a}}		 

log[a](b) = (1)/(log[b](a))

Log[a,b] == Divide[1,Log[b,a]]
Successful Successful - Successful [Tested: 36]
4.2.E11 e = 2.71828β€…18284β€…59045β€…23536 ⁒ … 𝑒 2.71828β€…18284β€…59045β€…23536 … {\displaystyle{\displaystyle e=2.71828\ 18284\ 59045\ 23536\dots}}
e = 2.71828\ 18284\ 59045\ 23536\dots		 

exp(1) = 2.71828182845904523536

E == 2.71828182845904523536
Successful Successful - Successful [Tested: 1]
4.2.E12 ln ⁑ e = 1 𝑒 1 {\displaystyle{\displaystyle\ln e=1}}
\ln@@{e} = 1		 

ln(exp(1)) = 1

Log[E] == 1
Successful Successful - Successful [Tested: 1]
4.2.E13 ∫ 1 e d t t = 1 superscript subscript 1 𝑒 𝑑 𝑑 1 {\displaystyle{\displaystyle\int_{1}^{e}\frac{\mathrm{d}t}{t}=1}}
\int_{1}^{e}\frac{\diff{t}}{t} = 1		 

int((1)/(t), t = 1..exp(1)) = 1

Integrate[Divide[1,t], {t, 1, E}, GenerateConditions->None] == 1
Successful Successful - Successful [Tested: 1]
4.2.E14 log e ⁑ z = ln ⁑ z 𝑒 𝑧 𝑧 {\displaystyle{\displaystyle\operatorname{log}_{e}z=\ln z}}
\genlog{e}@@{z} = \ln@@{z}		 

log[exp(1)](z) = ln(z)

Log[E,z] == Log[z]
Successful Successful - Successful [Tested: 7]
4.2.E15 log 10 ⁑ z = ( ln ⁑ z ) / ( ln ⁑ 10 ) 10 𝑧 𝑧 10 {\displaystyle{\displaystyle\operatorname{log}_{10}z=\ifrac{(\ln z)}{(\ln 10)}}}
\genlog{10}@@{z} = \ifrac{(\ln@@{z})}{(\ln@@{10})}		 

log[10](z) = (ln(z))/(ln(10))

Log[10,z] == Divide[Log[z],Log[10]]
Successful Successful - Successful [Tested: 7]
4.2.E15 ( ln ⁑ z ) / ( ln ⁑ 10 ) = ( log 10 ⁑ e ) ⁒ ln ⁑ z 𝑧 10 10 𝑒 𝑧 {\displaystyle{\displaystyle\ifrac{(\ln z)}{(\ln 10)}=(\operatorname{log}_{10}% e)\ln z}}
\ifrac{(\ln@@{z})}{(\ln@@{10})} = (\genlog{10}@@{e})\ln@@{z}		 

(ln(z))/(ln(10)) = (log[10](exp(1)))*ln(z)

Divide[Log[z],Log[10]] == (Log[10,E])*Log[z]
Successful Successful - Successful [Tested: 7]
4.2.E16 ln ⁑ z = ( ln ⁑ 10 ) ⁒ log 10 ⁑ z 𝑧 10 10 𝑧 {\displaystyle{\displaystyle\ln z=(\ln 10)\operatorname{log}_{10}z}}
\ln@@{z} = (\ln@@{10})\genlog{10}@@{z}		 

ln(z) = (ln(10))*log[10](z)

Log[z] == (Log[10])*Log[10,z]
Successful Successful - Successful [Tested: 7]
4.2.E17 log 10 ⁑ e = 0.43429β€…44819β€…03251β€…82765 ⁒ … 10 𝑒 0.43429β€…44819β€…03251β€…82765 … {\displaystyle{\displaystyle\operatorname{log}_{10}e=0.43429\ 44819\ 03251\ 82% 765\dots}}
\genlog{10}@@{e} = 0.43429\ 44819\ 03251\ 82765\dots		 

log[10](exp(1)) = 0.43429448190325182765

Log[10,E] == 0.43429448190325182765
Failure Successful Successful [Tested: 0] Successful [Tested: 1]
4.2.E18 ln ⁑ 10 = 2.30258β€…50929β€…94045β€…68401 ⁒ … 10 2.30258β€…50929β€…94045β€…68401 … {\displaystyle{\displaystyle\ln 10=2.30258\ 50929\ 94045\ 68401\dots}}
\ln@@{10} = 2.30258\ 50929\ 94045\ 68401\dots		 

ln(10) = 2.30258509299404568401

Log[10] == 2.30258509299404568401
Successful Successful - Successful [Tested: 1]
4.2.E20 exp ⁑ ( z + 2 ⁒ Ο€ ⁒ i ) = exp ⁑ z 𝑧 2 πœ‹ 𝑖 𝑧 {\displaystyle{\displaystyle\exp\left(z+2\pi i\right)=\exp z}}
\exp@{z+2\pi i} = \exp@@{z}		 

exp(z + 2*Pi*I) = exp(z)

Exp[z + 2*Pi*I] == Exp[z]
Successful Successful - Successful [Tested: 7]
4.2.E21 exp ⁑ ( - z ) = 1 / exp ⁑ ( z ) 𝑧 1 𝑧 {\displaystyle{\displaystyle\exp\left(-z\right)=1/\exp\left(z\right)}}
\exp@{-z} = 1/\exp@{z}		 

exp(- z) = 1/exp(z)

Exp[- z] == 1/Exp[z]
Successful Successful - Successful [Tested: 7]
4.2.E22 | exp ⁑ z | = exp ⁑ ( β„œ ⁑ z ) 𝑧 𝑧 {\displaystyle{\displaystyle|\exp z|=\exp\left(\Re z\right)}}
|\exp@@{z}| = \exp@{\realpart@@{z}}		 

abs(exp(z)) = exp(Re(z))

Abs[Exp[z]] == Exp[Re[z]]
Successful Successful - Successful [Tested: 7]
4.2.E23 ph ⁑ ( exp ⁑ z ) = β„‘ ⁑ z + 2 ⁒ k ⁒ Ο€ phase 𝑧 𝑧 2 π‘˜ πœ‹ {\displaystyle{\displaystyle\operatorname{ph}\left(\exp z\right)=\Im z+2k\pi}}
\phase@{\exp@@{z}} = \imagpart@@{z}+2k\pi		 

argument(exp(z)) = Im(z)+ 2*k*Pi

Arg[Exp[z]] == Im[z]+ 2*k*Pi
Failure Failure
Failed [21 / 21]
Result: -6.283185308
Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 1, k = 3}


Result: -12.56637062
Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 2, k = 3}


Result: -18.84955592
Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 3, k = 3}


Result: -6.283185308
Test Values: {z = -1/2+1/2*I*3^(1/2), k = 1, k = 3}

... skip entries to safe data
Failed [7 / 7]
Result: -18.84955592153876
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: -18.84955592153876
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.2.E24 exp ⁑ z = e x ⁒ cos ⁑ y + i ⁒ e x ⁒ sin ⁑ y 𝑧 superscript 𝑒 π‘₯ 𝑦 𝑖 superscript 𝑒 π‘₯ 𝑦 {\displaystyle{\displaystyle\exp z=e^{x}\cos y+ie^{x}\sin y}}
\exp@@{z} = e^{x}\cos@@{y}+ie^{x}\sin@@{y}		 

exp(x + y*I) = exp(x)*cos(y)+ I*exp(x)*sin(y)

Exp[x + y*I] == Exp[x]*Cos[y]+ I*Exp[x]*Sin[y]
Successful Successful - Successful [Tested: 18]
4.2.E26 z a = exp ⁑ ( a ⁒ Ln ⁑ z ) superscript 𝑧 π‘Ž π‘Ž multivalued-natural-logarithm 𝑧 {\displaystyle{\displaystyle z^{a}=\exp\left(a\operatorname{Ln}z\right)}}
z^{a} = \exp@{a\Ln@@{z}}		 z β‰  0 𝑧 0 {\displaystyle{\displaystyle z\neq 0}} 
(z)^(a) = exp(a*ln(z))

(z)^(a) == Exp[a*Log[z]]
Successful Successful - Successful [Tested: 42]
4.2.E28 z a = exp ⁑ ( a ⁒ ln ⁑ z ) superscript 𝑧 π‘Ž π‘Ž 𝑧 {\displaystyle{\displaystyle z^{a}=\exp\left(a\ln z\right)}}
z^{a} = \exp@{a\ln@@{z}}		 

(z)^(a) = exp(a*ln(z))

(z)^(a) == Exp[a*Log[z]]
Successful Successful - Successful [Tested: 42]
4.2.E29 | z a | = | z | β„œ ⁑ a ⁒ exp ⁑ ( - ( β„‘ ⁑ a ) ⁒ ph ⁑ z ) superscript 𝑧 π‘Ž superscript 𝑧 π‘Ž π‘Ž phase 𝑧 {\displaystyle{\displaystyle|z^{a}|=|z|^{\Re a}\exp\left(-(\Im a)\operatorname% {ph}z\right)}}
|z^{a}| = |z|^{\realpart@@{a}}\exp@{-(\imagpart@@{a})\phase@@{z}}		 

abs((z)^(a)) = (abs(z))^(Re(a))* exp(-(Im(a))*argument(z))

Abs[(z)^(a)] == (Abs[z])^(Re[a])* Exp[-(Im[a])*Arg[z]]
Failure Failure Successful [Tested: 42] Successful [Tested: 42]
4.2.E30 ph ⁑ ( z a ) = ( β„œ ⁑ a ) ⁒ ph ⁑ z + ( β„‘ ⁑ a ) ⁒ ln ⁑ | z | phase superscript 𝑧 π‘Ž π‘Ž phase 𝑧 π‘Ž 𝑧 {\displaystyle{\displaystyle\operatorname{ph}\left(z^{a}\right)=(\Re a)% \operatorname{ph}z+(\Im a)\ln|z|}}
\phase@{z^{a}} = (\realpart@@{a})\phase@@{z}+(\imagpart@@{a})\ln@@{|z|}		 

argument((z)^(a)) = (Re(a))*argument(z)+(Im(a))*ln(abs(z))

Arg[(z)^(a)] == (Re[a])*Arg[z]+(Im[a])*Log[Abs[z]]
Failure Failure
Failed [6 / 42]
Result: -6.283185308
Test Values: {a = -1.5, z = -1/2*3^(1/2)-1/2*I}


Result: 6.283185308
Test Values: {a = 1.5, z = -1/2*3^(1/2)-1/2*I}


Result: 6.283185307
Test Values: {a = -2, z = -1/2+1/2*I*3^(1/2)}


Result: -6.283185309
Test Values: {a = -2, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [6 / 42]
Result: -6.283185307179586
Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

Result: 6.283185307179586
Test Values: {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

... skip entries to safe data
4.2#Ex1 | z a | = | z | a superscript 𝑧 π‘Ž superscript 𝑧 π‘Ž {\displaystyle{\displaystyle|z^{a}|=|z|^{a}}}
|z^{a}| = |z|^{a}		 

abs((z)^(a)) = (abs(z))^(a)
 
Abs[(z)^(a)] == (Abs[z])^(a)
 Skipped - no semantic math Skipped - no semantic math - -
4.2#Ex2 ph ⁑ ( z a ) = a ⁒ ph ⁑ z phase superscript 𝑧 π‘Ž π‘Ž phase 𝑧 {\displaystyle{\displaystyle\operatorname{ph}\left(z^{a}\right)=a\operatorname% {ph}z}}
\phase@{z^{a}} = a\phase@@{z}		 

argument((z)^(a)) = a*argument(z)
 
Arg[(z)^(a)] == a*Arg[z]
 Failure Failure
Failed [6 / 42]
Result: -6.283185308
Test Values: {a = -1.5, z = -1/2*3^(1/2)-1/2*I}

Result: 6.283185308
Test Values: {a = 1.5, z = -1/2*3^(1/2)-1/2*I}

Result: 6.283185307
Test Values: {a = -2, z = -1/2+1/2*I*3^(1/2)}

Result: -6.283185309
Test Values: {a = -2, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [6 / 42]
Result: -6.283185307179586
Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

Result: 6.283185307179586
Test Values: {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

... skip entries to safe data
4.2.E32 e z = exp ⁑ z superscript 𝑒 𝑧 𝑧 {\displaystyle{\displaystyle e^{z}=\exp z}}
e^{z} = \exp@@{z}		 

exp(z) = exp(z)
 
Exp[z] == Exp[z]
 Successful Successful - Successful [Tested: 7]
4.2.E33 e z = ( exp ⁑ z ) ⁒ exp ⁑ ( 2 ⁒ k ⁒ z ⁒ Ο€ ⁒ i ) superscript 𝑒 𝑧 𝑧 2 π‘˜ 𝑧 πœ‹ imaginary-unit {\displaystyle{\displaystyle e^{z}=(\exp z)\exp\left(2kz\pi\mathrm{i}\right)}}
e^{z} = (\exp@@{z})\exp@{2kz\pi\iunit}		 

exp(z) = (exp(z))*exp(2*k*z*Pi*I)
 
Exp[z] == (Exp[z])*Exp[2*k*z*Pi*I]
 Failure Failure
Failed [16 / 21]
Result: 1.989606315+1.174241786*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 1, k = 3}

Result: 2.084725711+1.143917762*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 2, k = 3}

Result: 2.086486474+1.139979111*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 3, k = 3}

Result: .3946493584+.4640329579*I
Test Values: {z = -1/2+1/2*I*3^(1/2), k = 1, k = 3}

... skip entries to safe data
Failed [6 / 7]
Result: Complex[2.0864864733305994, 1.139979110702337]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[0.3929465878104918, 0.4620308216689905]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.2.E36 - Ο€ ≀ β„‘ ⁑ ( 1 a ⁒ Ln ⁑ w ) πœ‹ 1 π‘Ž multivalued-natural-logarithm 𝑀 {\displaystyle{\displaystyle-\pi\leq\Im\left(\frac{1}{a}\operatorname{Ln}w% \right)}}
-\pi \leq \imagpart@@{\left(\frac{1}{a}\Ln@@{w}\right)}		 

- Pi <= Im((1)/(a)*ln(w))
 
- Pi <= Im[Divide[1,a]*Log[w]]
 Failure Failure
Failed [5 / 60]
Result: -3.141592654 <= -4.188790204
Test Values: {a = -.5, w = -1/2+1/2*I*3^(1/2)}

Result: -3.141592654 <= -6.283185308
Test Values: {a = -.5, w = -1.5}

Result: -3.141592654 <= -6.283185308
Test Values: {a = -.5, w = -.5}

Result: -3.141592654 <= -6.283185308
Test Values: {a = -.5, w = -2}

... skip entries to safe data
Failed [5 / 60]
Result: False
Test Values: {Rule[a, -0.5], Rule[w, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

Result: False
Test Values: {Rule[a, -0.5], Rule[w, -1.5]}

... skip entries to safe data
4.2.E36 β„‘ ⁑ ( 1 a ⁒ Ln ⁑ w ) ≀ Ο€ 1 π‘Ž multivalued-natural-logarithm 𝑀 πœ‹ {\displaystyle{\displaystyle\Im\left(\frac{1}{a}\operatorname{Ln}w\right)\leq% \pi}}
\imagpart@@{\left(\frac{1}{a}\Ln@@{w}\right)} \leq \pi		 

Im((1)/(a)*ln(w)) <= Pi
 
Im[Divide[1,a]*Log[w]] <= Pi
 Failure Failure
Failed [5 / 60]
Result: 5.235987758 <= 3.141592654
Test Values: {a = -.5, w = -1/2*3^(1/2)-1/2*I}

Result: 4.188790204 <= 3.141592654
Test Values: {a = .5, w = -1/2+1/2*I*3^(1/2)}

Result: 6.283185308 <= 3.141592654
Test Values: {a = .5, w = -1.5}

Result: 6.283185308 <= 3.141592654
Test Values: {a = .5, w = -.5}

... skip entries to safe data
Failed [5 / 60]
Result: False
Test Values: {Rule[a, -0.5], Rule[w, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

Result: False
Test Values: {Rule[a, 0.5], Rule[w, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
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