Cosine Integral

Author

Eric W. Weisstein
June 20, 2004

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

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Definition

-Integrate[((Cos[t])/t),{t,x,∞},Assumptions->x>0]

EulerGamma+Log[z]+Integrate[((Cos[t]-1)/t),{t,0,z}]

FullSimplify[(1/2)(ExpIntegralEi[I x]+ExpIntegralEi[-I x]),x>0]

Plots

<<Utilities`Plot`

Real Axis

Plot[CosIntegral[x],{x,0,10},PlotRange->{Automatic,{-4,.5}},
PlotStyle->Red,
AxesLabel->TraditionalForm/@{x,CosIntegral[x]}]

Plot in Complex Plane

ComplexPlot3D[CosIntegral[z],{z,-2Pi,2Pi}]//Timing

ComplexContourPlot[CosIntegral[z],{z,-2Pi,2Pi}]//Timing

Derivative

D[CosIntegral[z],z]

Integral

Integrate[CosIntegral[z],z]

Series

(∑_k^∞)((((-(x^2)))^k)/(2 k (2 k)!))+ln(x)+ℽ==Ci(x)

V6

EulerGamma+Log[x]+Sum[((((-(x^2)))^k)/(2k(2k)!)),{k,∞}]

V7

EulerGamma+Log[x]+Sum[((((-(x^2)))^k)/(2k(2k)!)),{k,∞}]

Roots

v4.2

Reduce[CosIntegral[x]==0&&x∈Reals,x]

v5.0

Reduce[CosIntegral[x]==0&&x∈Reals,x]

Numerical

FindRoot[CosIntegral[x]==0,{x,.2}]

FindRoot[CosIntegral[x]==0,{x,3.8}]

FindRoot[CosIntegral[x]==0,{x,6}]

D[CosIntegral[x],x]==0

Table[CosIntegral[n Pi/2.],{n,1,5,2}]

D[CosIntegral[x],{x,2}]

FindRoot[-((Cos[x])/(x^2))-((Sin[x])/x)==0,{x,1}]

FindRoot[-((Cos[x])/(x^2))-((Sin[x])/x)==0,{x,5}]

FindRoot[-((Cos[x])/(x^2))-((Sin[x])/x)==0,{x,8}]

Integrals

∫(x^(2 n)) cos(m x)dx

This is wrong

Sum

FullSimplify[(((-1))^(n+1))(2n)!(Sin[m x]Sum[((((-1))^k)/((2k)!(m^(2n-2k+1))))(x^(2k)),{k,0,n}]+Cos[m x]Sum[((((-1))^(k+1))/((2k-3)!(m^(2n-2k+2))))(x^(2k-1)),{k,n}]),m>0&&n>0&&n∈Integers&&x>0]

Subtract@@@Table[{(((-1))^(n+1))(2n)!(Sin[m x]Sum[((((-1))^k)/((2k)!(m^(2n-2k+1))))(x^(2k)),{k,0,n}]+Cos[m x]Sum[((((-1))^(k+1))/((2k-3)!(m^(2n-2k+2))))(x^(2k-1)),{k,n}]),Integrate[(x^(2n))Cos[m x],x]},{n,2},{m,2}]//FullSimplify

v5.0.1

FullSimplify[Integrate[(x^(2n))Cos[m x],x,Assumptions->m>0&&n>0&&n∈Integers],
m>0&&n>0&&n∈Integers]

v5.1

FullSimplify[Integrate[(x^(2n))Cos[m x],x,Assumptions->m>0&&n>0&&n∈Integers],
m>0&&n>0&&n∈Integers]

∫(cos^m)(x)dx

FullSimplify[Integrate[((Cos[x])^m),x]]

Subtract@@@Table[{
If[EvenQ[m],(((2(m/2)-1)!!)/((2(m/2))!!))(Sin[x]Sum[(((2k)!!)/((2k+1)!!))((Cos[x])^(2k+1)),{k,0,(m/2)-1}]+x),
(((2((m-1)/2))!!)/((2((m-1)/2)+1)!!))Sin[x]Sum[(((2k-1)!!)/((2k)!!))((Cos[x])^(2k)),{k,0,((m-1)/2)}]],
Integrate[((Cos[x])^m),x]},{m,10}]//Simplify