({{, (RefLink[ComposeSeries, paclet:ref/ComposeSeries])[(Subscript[series, 1]),(Subscript[series, 2]),…]
composes several power series. }})

OpenersNotesSection(Spacer1)Details

(RefLink[ComposeSeries, paclet:ref/ComposeSeries])[(Subscript[series, 1]),(Subscript[series, 2]),…] effectively replaces the variable in (Subscript[series, 1]) by (Subscript[series, 2]) and so on.

Two series can only meaningfully be composed when the point about which the first series is expanded corresponds to the limiting value of the second series at its expansion point.

OpenersPrimaryExamplesSection(Spacer1)Examples  (5)

OpenersExampleSection0(Spacer1)Basic Examples  (2)

Compose two power series:

ComposeSeries[x+2x^2+O[x]^12,3 x+4x^2+O[x]^10]

ComposeSeries[Series[Exp[x],{x,0,10}],Series[Sin[x],{x,0,10}]]

Series[Exp[Sin[x]],{x,0,10}]

OpenersExampleSection1(Spacer1)Properties & Relations  (2)

(RefLink[ComposeSeries, paclet:ref/ComposeSeries]) generates the series of composition of functions:

f[x_]:=x Exp[x];
g[x_]:=Sin[x];

sf=Series[f[x],{x,0,10}];
sg=Series[g[x],{x,0,10}];

ComposeSeries[sf,sg]

Series[f[g[x]],{x,0,10}]

Coefficients of the composition of series can be computed using (RefLink[BellY, paclet:ref/BellY]):

Sum[BellY[Table[{i! (Subscript[a, i]),i! (Subscript[b, i])},{i,n}]]x^n/n!,{n,1,3}]+O[x]^4

ComposeSeries[(Subscript[a, 1]) x+(Subscript[a, 2])x^2+(Subscript[a, 3]) x^3+O[x]^4,(Subscript[b, 1]) x+(Subscript[b, 2]) x^2+(Subscript[b, 3]) x^3+O[x]^4]

OpenersExampleSection2(Spacer1)Possible Issues  (1)

The limiting value of the second series must coincide with the expansion point of the first series:

ComposeSeries[x+2x^2+O[x]^10,1+3x+O[x]^5]

ComposeSeries[Series[Log[x],{x,1,5}],1+3x+O[x]^5]

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[Series, paclet:ref/Series])  ▪  (RefLink[InverseSeries, paclet:ref/InverseSeries])  ▪  (RefLink[SeriesData, paclet:ref/SeriesData])}})}})

({{DocumentationFooterIconsRelatedTutorial.png, ({{Tutorials}, {▪(RefLinkPlain[Power Series, paclet:tutorial/PowerSeries])}, {▪(RefLinkPlain[Making Power Series Expansions, paclet:tutorial/MakingPowerSeriesExpansions])}, {▪(RefLinkPlain[Operations on Power Series, paclet:tutorial/OperationsOnPowerSeries])}, {▪(RefLinkPlain[The Representation of Power Series, paclet:tutorial/TheRepresentationOfPowerSeries])}, {▪(RefLinkPlain[Composition and Inversion of Power Series, paclet:tutorial/CompositionAndInversionOfPowerSeries])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Series Expansions, paclet:guide/SeriesExpansions])}})}})