Inverse Gudermannian

Author

Eric W. Weisstein
September 20, 2008

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

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

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

Notation

InverseGudermannian[x]//TraditionalForm

Identities

(gd^(-1))(x)==log(tan((x/2)+(π/4)))

InverseGudermannian[x]//FunctionExpand//TraditionalForm

(∫_0^x)sectdt==(gd^(-1))(x)

Integrate[Sec[t],{t,0,x},Assumptions->-Pi/2≤x<Pi/2]

FullSimplify[%]

log(cos((x/2))+sin((x/2)))-log(cos((x/2))-sin((x/2)))

diff=FullSimplify[Log[Cos[x/2]+Sin[x/2]]-Log[Cos[x/2]-Sin[x/2]]-InverseGudermannian[x],-Pi/2≤x<Pi/2]

2 (tanh^(-1))(tan((x/2)))

FullSimplify[2 ArcTanh[Tan[(x/2)]]-InverseGudermannian[x],-Pi/2≤x<Pi/2]

diff=FullSimplify[(2 ArcTanh[Tan[(x/2)]]-InverseGudermannian[x])//FunctionExpand,-Pi/2≤x<Pi/2]

Plot[diff,{x,-Pi/2,Pi/2}]

(1/2) log(((sin(x)+1)/(1-sin(x))))

diff=FullSimplify[Subtract@@{InverseGudermannian[x],(1/2)Log[((1+Sin[x])/(1-Sin[x]))]},0<x<π/2]

Plot[diff,{x,-π/2,π/2}]

(1/2) log(((sin(x)+1)/(1-sin(x))))

diff=FullSimplify[Subtract@@{InverseGudermannian[x],Log[Sec[x]+Tan[x]]},0<x<π/2]

log(tan((x/2)+(π/4)))

diff=FullSimplify[Subtract@@{InverseGudermannian[x],Log[Tan[Pi/4+x/2]]},-π/2<x<π/2]

log(sec(x)+tan(x))

diff=FullSimplify[Subtract@@{InverseGudermannian[x],Log[Sec[x]+Tan[x]]},0<x<π/2]

Plot[diff,{x,-π/2,π/2}]

Derivative and Integral

Derivative

D[InverseGudermannian[x],x]

Integrate[%,x]

FullSimplify[%]

<<Utilities`Plot`

Plot[Through[{Re,Im}[-Log[Cos[(x/2)]-Sin[(x/2)]]+Log[Cos[(x/2)]+Sin[(x/2)]]-InverseGudermannian[x]]],{x,-20,20},WorkingPrecision->30,Evaluated->True]

-Log[Cos[(x/2)]-Sin[(x/2)]]+Log[Cos[(x/2)]+Sin[(x/2)]]-InverseGudermannian[x]/.x->-9.`20

Integral

Integrate[InverseGudermannian[x],x]//FullSimplify//TraditionalForm

Integrate[InverseGudermannian[x],x]

FullSimplify[%]

%/.Tan[(1/4) (π+2 x)]->Exp[InverseGudermannian[x]]//Simplify

Subtract@@@Table[{Tan[(1/4) (π+2 x)],Exp[InverseGudermannian[x]]}/.x->RandomComplex[],{10}]//Chop//Union

D[%,x]

FullSimplify[Log[Tan[(1/4) (π+2 x)]]]

FullSimplify[Log[Tan[(1/4) (π+2 x)]]-InverseGudermannian[x]]

Series

ser=Series[InverseGudermannian[x],{x,0,15}]

ser=CoefficientList[%,x]

Through[{Numerator,Denominator}[ser[[2;;-1;;2]]]]

SeriesCoefficient[InverseGudermannian[x],{x,0,n}]

Series[Log[Tan[x]+Sec[x]],{x,0,20}]

TimeConstrained[SeriesCoefficient[Log[Tan[x]+Sec[x]],{x,0,n}],60]//Timing