Hyperbolic Function

Author

Eric W. Weisstein
November 25, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/HyperbolicFunctions.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/HyperbolicFunctions.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Identities

FactorTerms[TrigToExp[Cos[z]]]

FactorTerms[TrigToExp[Cosh[z]]]

sinh(i x)==i sin(x)

FunctionExpand[Sinh[I x]]

cosh(i x)==cos(x)

FunctionExpand[Cosh[I x]]

Osborne's rule analogs

FullSimplify[((Cosh[x])^2)-((Sinh[x])^2)]

FullSimplify[Cosh[x]+Sinh[x]]

FullSimplify[Cosh[x]-Sinh[x]]

Half-Angle Formula

ComplexExpand[Tanh[((x+I y)/2)]]//Together

ComplexExpand[Coth[((x+I y)/2)]]//Together//ExpandNumerator

Double-Angle Formulas

TrigExpand[Sinh[2z]]

TrigExpand[Cosh[2z]]

Complex Arguments

TrigExpand[Sinh[x+I z]]

TrigExpand[Cosh[x+I z]]

Absolute Squares

TrigExpand[ComplexExpand[Abs[TrigExpand[Sinh[x+I y]]]]^2//FullSimplify]//FullSimplify

FullSimplify[((Sinh[x])^2)+((Sin[y])^2)-(1/2) (-Cos[2 y]+Cosh[2 x])]

ByteCount/@{(1/2) (-Cos[2 y]+Cosh[2 x]),((Sinh[x])^2)+((Sin[y])^2)}

TrigExpand[ComplexExpand[Abs[TrigExpand[Cosh[x+I y]]]]^2//FullSimplify]//FullSimplify

FullSimplify[Sinh[x]^2+Cos[y]^2-%]

ByteCount/@{(1/2) (Cos[2 y]+Cosh[2 x]),Sinh[x]^2+Cos[y]^2}

Integrals

∫(1/(x (Sqrt[a+b x])))dx==-(2/(Sqrt[a])) (tanh^(-1))(((Sqrt[a+b x])/(Sqrt[a])))==log((((Sqrt[a+b x])-(Sqrt[a]))/((Sqrt[a+b x])+(Sqrt[a]))))

i=Integrate[(1/(x(Sqrt[a+b x]))),x]

Plot[Evaluate[Through[{Re,Im}[i/.{a->1,b->2}]]],{x,-2,2},PlotStyle->Thickness[.01]]

Plot[Evaluate[Through[{Re,Im}[Log[(((Sqrt[a+b x])-(Sqrt[a]))/((Sqrt[a+b x])+(Sqrt[a])))]/.{a->1,b->2}]]],{x,-2,2},PlotStyle->Thickness[.01]]

Plot3D[Evaluate[Through[{Re,Im}[-(2/(Sqrt[a])) ArcTanh[((Sqrt[a+b (x+I y)])/(Sqrt[a]))]-Log[(((Sqrt[a+b (x+I y)])-(Sqrt[a]))/((Sqrt[a+b (x+I y)])+(Sqrt[a])))]/.{a->1,b->2}]]],{x,-5,5},{y,-5,5},PlotStyle->{Red,Yellow}]

(x|a|b)∈ℂ

Table[Log[Abs[(((Sqrt[a+b x])-(Sqrt[a]))/((Sqrt[a+b x])+(Sqrt[a])))]]-Log[Abs[(((a+b x)-2(Sqrt[a(a+b x)])+a)/(b x))]]/.{x->RandomComplexDisk[1,1WorkingPrecision->20],a->RandomComplexDisk[1,1WorkingPrecision->20],b->RandomComplexDisk[1,1WorkingPrecision->20]},{10}]//Chop//Union

(x|b)∈ℂ,a>0

Table[Log[(((Sqrt[a+b x])-(Sqrt[a]))/((Sqrt[a+b x])+(Sqrt[a])))]-Log[(((2a+b x)-2(Sqrt[a(a+b x)]))/(b x))]/.{x->RandomComplexDisk[1,1,WorkingPrecision->20],a->RandomReal[{0,1}],b->RandomComplexDisk[1,1,WorkingPrecision->20]},{1000}]//Chop//Union

(x|a)∈ℂ,b∈Reals

Table[Log[(((Sqrt[a+b x])-(Sqrt[a]))/((Sqrt[a+b x])+(Sqrt[a])))]-Log[(((2a+b x)-2(Sqrt[a(a+b x)]))/(b x))]/.{x->RandomComplexDisk[1,1,WorkingPrecision->20],b->RandomReal[WorkingPrecision->20],a->RandomComplexDisk[1,1,WorkingPrecision->20]},{10}]//Chop//Union

(x|a)∈ℂ,b>0

Table[Log[(((Sqrt[a+b x])-(Sqrt[a]))/((Sqrt[a+b x])+(Sqrt[a])))]-Log[(((2a+b x)-2(Sqrt[a(a+b x)]))/(b x))]/.{x->RandomComplexDisk[1,1,WorkingPrecision->20],b->RandomReal[{0,1},WorkingPrecision->20],a->RandomComplexDisk[1,1,WorkingPrecision->20]},{10}]//Chop//Union

x∈ℂ,a,b>0

Table[Log[(((Sqrt[a+b x])-(Sqrt[a]))/((Sqrt[a+b x])+(Sqrt[a])))]-Log[(((2a+b x)-2(Sqrt[a(a+b x)]))/(b x))]/.{x->RandomComplexDisk[1,1,WorkingPrecision->20],b->RandomReal[{0,1},WorkingPrecision->20],a->RandomReal[{0,1},WorkingPrecision->20]},{10}]//Chop//Union

Identity

Table[Log[(((Sqrt[a+b x])-(Sqrt[a]))/((Sqrt[a+b x])+(Sqrt[a])))]-Log[(((2a)/(b x))+1)-(Sqrt[(((((2a)/(b x))+1))^2)+1])]/.{x->RandomComplexDisk[1,1,WorkingPrecision->20],b->RandomReal[{0,1},WorkingPrecision->20],a->RandomReal[{0,1},WorkingPrecision->20]},{10}]//Chop//Union

Table[Log[(((Sqrt[a+b x])-(Sqrt[a]))/((Sqrt[a+b x])+(Sqrt[a])))]-ArcSinh[1+((2a)/(b x))]/.{x->RandomComplexDisk[1,1,WorkingPrecision->20],b->RandomReal[{0,1},WorkingPrecision->20],a->RandomReal[{0,1},WorkingPrecision->20]},{10}]//Chop//Union

Table[Log[(((Sqrt[a+b x])-(Sqrt[a]))/((Sqrt[a+b x])+(Sqrt[a])))]-2ArcTanh[-(Sqrt[(a/(a+b x))])]/.{x->RandomComplexDisk[1,1,WorkingPrecision->20],b->RandomReal[{0,1},WorkingPrecision->20],a->RandomReal[{0,1},WorkingPrecision->20]},{20}]//Chop//Union

Table[Log[(((Sqrt[a+b x])-(Sqrt[a]))/((Sqrt[a+b x])+(Sqrt[a])))]+(2/(Sqrt[a])) ArcTanh[((Sqrt[a+b x])/(Sqrt[a]))]/.{x->RandomReal[{0,1},WorkingPrecision->20],b->RandomReal[{0,1},WorkingPrecision->20],a->RandomReal[{0,1},WorkingPrecision->20]},{20}]//Chop//Union

Table[Log[(((Sqrt[a+b x])-(Sqrt[a]))/((Sqrt[a+b x])+(Sqrt[a])))]+(2/(Sqrt[a])) ArcTanh[((Sqrt[a+b x])/(Sqrt[a]))]/.{x->RandomComplexDisk[1,1,WorkingPrecision->20],b->RandomReal[{0,1},WorkingPrecision->20],a->RandomReal[{0,1},WorkingPrecision->20]},{20}]//Chop//Union

Log[z-(Sqrt[(z^2)-1])]-ArcCosh[z]/.z->.2

-((2 ArcTanh[((Sqrt[a+b x])/(Sqrt[a]))])/(Sqrt[a]))