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Series Expansion


A series expansion is a representation of a particular function as a sum of powers in one of its variables, or by a sum of powers of another (usually elementary) function f(x).

Here are series expansions (some Maclaurin, some Laurent, and some Puiseux) for a number of common functions.

1/(1-x)=1+x+x^2+x^3+x^4+x^5+... for -1<x<1
(1)
cn(x,k)=1-1/2x^2+1/(24)(1+4k^2)x^4+...
(2)
cosx=1-1/2x^2+1/(24)x^4-1/(720)x^6-... for -infty<x<infty
(3)
cos^(-1)x=1/2pi-x-1/6x^3-3/(40)x^5-5/(112)x^7-... for -1<x<1
(4)
coshx=1+1/2x^2+1/(24)x^4+1/(720)x^6+1/(40,320)x^8+...
(5)
cosh^(-1)(1+x)=sqrt(2x)(1-1/(12)x+3/(160)x^2-5/(896)x^3+...)
(6)
cotx=x^(-1)-1/3x-1/(45)x^3-2/(945)x^5-1/(4725)x^7-...
(7)
cot^(-1)x=1/2pi-x+1/3x^3-1/5x^5+1/7x^7-1/9x^9+...
(8)
cot^(-1)(1/x)=x-1/3x^3+1/5x^5-1/7x^7+1/9x^9+...
(9)
cothx=x^(-1)+1/3x-1/(45)x^3+2/(945)x^5-1/(4725)x^7+...
(10)
coth^(-1)(1+x)=1/2ln2-1/2lnx+1/4x-1/(16)x^2+...
(11)
cscx=x^(-1)+1/6x+7/(360)x^3+(31)/(15120)x^5+...
(12)
cschx=x^(-1)-1/6x+7/(360)x^3+(31)/(15120)x^5+...
(13)
csch^(-1)x=ln2-lnx+1/4x^2-3/(32)x^4+5/(96)x^6-...
(14)
dn(x,k)=1-1/2k^2x^2+1/(24)k^2(4+k^2)x^4+...
(15)
erfx=1/(sqrt(pi))(2x-2/3x^3+1/5x^5-1/(21)x^7+...)
(16)
e^x=1+x+1/2x^2+1/6x^3+1/(24)x^4+... for -infty<x<infty
(17)
_2F_1(alpha,beta,gamma;x)=1+(alphabeta)/(1!gamma)x+(alpha(alpha+1)beta(beta+1))/(2!gamma(gamma+1))x^2+...
(18)
ln(1+x)=x-1/2x^2+1/3x^3-1/4x^4+... for -1<x<1
(19)
ln((1+x)/(1-x))=2x+2/3x^3+2/5x^5+2/7x^7+... for -1<x<1
(20)
secx=1+1/2x^2+5/(24)x^4+(61)/(720)x^6+(277)/(8064)x^8+...
(21)
sechx=1-1/2x^2+5/(24)x^4-(61)/(720)x^6+(277)/(8064)x^8+...
(22)
sech^(-1)x=ln2-lnx-1/4x^2-3/(32)x^4-...
(23)
sinx=x-1/6x^3+1/(120)x^5-1/(5040)x^7+... for -infty<x<infty
(24)
sin^(-1)x=x+1/6x^3+3/(40)x^5+5/(112)x^7+(35)/(1152)x^9+...
(25)
sinhx=x+1/6x^3+1/(120)x^5+1/(5040)x^7+1/(362880)x^9+...
(26)
sinh^(-1)x=x-1/6x^3+3/(40)x^5-5/(112)x^7+(35)/(1152)x^9-...
(27)
sn(x,k)=x-1/6(1+k^2)x^3+1/(120)(1+14k^2+k^4)x^5+...
(28)
tanx=x+1/3x^3+2/(15)x^5+(17)/(315)x^7+(62)/(2835)x^9+...
(29)
tan^(-1)x=x-1/3x^3+1/5x^5-1/7x^7+... for -1<x<1
(30)
tan^(-1)(1+x)=1/4pi+1/2x-1/4x^2+1/(12)x^3+1/(40)x^5+...
(31)
tanhx=x-1/3x^3+2/(15)x^5-(17)/(315)x^7+(62)/(2835)x^9+...
(32)
tanh^(-1)x=x+1/3x^3+1/5x^5+1/7x^7+1/9x^9+....
(33)

See also

Laurent Series, Maclaurin Series, Power Series, Puiseux Series, Series, Series Reversion, Taylor Series

This entry contributed by Dan Uznanski

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Cite this as:

Uznanski, Dan. "Series Expansion." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/SeriesExpansion.html

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