Lambda Function

Author

Eric W. Weisstein
October 6, 2008

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

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

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

Initialization

<<Utilities`Plot`

DirichletLambda

Plot[DirichletLambda[x],{x,0,2},Exclusions->1,PlotRange->10,AxesLabel->{x,HoldForm[DirichletLambda[x]]},Epilog->{Blue,Dashing[{.02}],Line[{{1,-100},{1,100}}]}]

Carmichael

DiscretePlot[CarmichaelLambda[n],{n,500},Joined->True,AxesLabel->{n,CarmichaelLambda[n]}]

ListPlot[Table[CarmichaelLambda[n],{n,0,500}],Joined->True,AxesLabel->{n,CarmichaelLambda[n]}]

Mangolt

ListPlot[Table[MangoldtLambda[n],{n,0,500}],Joined->True,AxesLabel->{n,MangoldtLambda[n]}]

Triangle Function

<<Utilities`Plot`

Plot[UnitTriangle[x],{x,-2,2},AxesLabel->{x,Λ[x]},PlotStyle->{Red,Thick}]

Jahnke and Emden

Lambda[n_,x_?NumericQ]:=Gamma[n+1]BesselJ[n,x]/(x/2)^n

FullSimplify[Gamma[n+1]BesselJ[n,x]/(x/2)^n /.n->0]

FullSimplify[Gamma[n+1]BesselJ[n,x]/(x/2)^n /.n->1]

V5

Plot[Evaluate[Table[Lambda[n,x],{n,0,4}]],{x,-15,15},PlotStyle->{Black,Red,Yellow,Green,Blue},PlotRange->All,AxesLabel->TraditionalForm/@{x,(Subscript[Λ, n])[x]}]

V7

Plot[Table[Lambda[n,x],{n,0,4}],{x,-15,15},PlotStyle->{Black,Red,Yellow,Green,Blue},PlotRange->All,AxesLabel->{x,(Subscript[Λ, n])[x]},Evaluated->True]

Two-Parameter

Integrate[((Gamma[t+1])/(x^t)),{t,0,y}]

Integrate[((Gamma[t+1])/(x^t)),{t,0,y},Assumptions->y>0&&x>0]

Integrate[((Gamma[t+1])/(x^t)),t]