InverseSeries
takes the series s, and gives a series for the inverse of the function represented by s.
InverseSeries[s,x]
uses the variable x in the inverse series.
Details
- InverseSeries performs "reversion" of series.
- Given a series
, InverseSeries[s,x] gives a series for 
such that 
. - InverseSeries can be applied to any SeriesData object with the appropriate structure, whether or not it has been generated by Series.
Examples
open all close allBasic Examples (3)
Find a series for the inverse function of x Sin[x]:
InverseSeries[Series[x Sin[x], {x, 0, 10}]]Find a series for the inverse function of Sin[x]:
InverseSeries[Series[Sin[x], {x, 0, 10}]]Series[ArcSin[x], {x, 0, 10}]Find the series for the inverse of the function represented by an explicitly specified series:
InverseSeries[x ^ 2 + 3x ^ 4 + O[x] ^ 8]Invert again to get back the original function:
InverseSeries[%]Generalizations & Extensions (1)
Applications (2)
Find higher-order terms in Newton's approximation for a root of f[x] near 
:
InverseSeries[Series[f[x], {x, a, 3}]]Normal[%] /. x -> 0Compute series expansion of 
at the origin, given 
:
serf = x + c1 x ^ 2 + c2 x ^ 3 + c3 x ^ 4 + c4 x ^ 5 + O[x] ^ 6;
serg = x + d1 x ^ 2 + d2 x ^ 3 + d3 x ^ 4 + d4 x ^ 5 + O[x] ^ 6;serCommute[s1_, s2_] := ComposeSeries[s1, s2] - ComposeSeries[s2, s1]serCommute[serf, serg] / serCommute[InverseSeries[serf], InverseSeries[serg]]//SimplifyText
Wolfram Research (1988), InverseSeries, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseSeries.html.
CMS
Wolfram Language. 1988. "InverseSeries." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/InverseSeries.html.
APA
Wolfram Language. (1988). InverseSeries. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/InverseSeries.html