SeriesCoefficient

The coefficient of $(x-x_0)^n$ in the Taylor series of $f$ about $x_0$, via $\frac{f^{(n)}(x_0)}{n!}$. Provided by @enumeratio/analytic.

✓ checked against that system · ! checked, with disagreements (hover a chip)FungrimSeriesCoefficient↗Wolfram LanguageSeriesCoefficient!↗

SeriesCoefficient(f, {x, x0, n})the coefficient of $(x-x_0)^n$, for $n$ a nonnegative integer and $f$ analytic at $x_0$ (an ordinary point).

Typed signatures

Type	Package	Applies
(expression, list<any^3>) -> number	analytic	always

Domain: Analytic

Details

• Exact for the ordinary-point Taylor case: $n$ nonnegative integer, $f$ with no pole or branch point at $x_0$. Computed as $D^n(f)$ evaluated at $x_0$, divided by $n!$ — no series expansion is built, just repeated differentiation.
• Also: $c\,x^m I_\nu(kx)$ and $c\,x^m J_\nu(kx)$ about $0$ ($\nu$ a nonnegative integer) at any index, symbolic $n$ included, read off the power series: a Piecewise that is zero unless $n-m-\nu$ is a nonnegative even number.
• Also: $c\,(bx+d)^a$ (a power, a root, a reciprocal) at a symbolic $n$, read off the binomial series: $c\binom{a}{n} b^n u_0^{a-n}$ with $u_0$ the base at $x_0$, a Piecewise that is zero for negative $n$. A real root (CubeRoot, an odd Root) keeps its real branch at a negative base. Wolfram writes the same sequence as a DifferenceRoot for a root.
• Declined on purpose: $n$ negative or non-integer (a Laurent or Puiseux coefficient — the derivative formula doesn't apply past an ordinary point) and $f$ singular exactly at $x_0$ even where the singularity is removable (e.g. $\sin(x)/x$ at $0$) — the result is checked for a leftover free occurrence of the expansion variable or a non-finite value, and declined rather than trusted, whenever either shows up.

Examples

1 more kept as data, checked against the oracles

Basic #

$[x^2]\, e^x$ about $x_0=1$ is $e/2$

live ↗

In$\mathrm{SeriesCoefficient}(e^{x}, \bigl\lbrack x, 1, 2\bigr\rbrack)$Out$\frac{e}{2}$

$[x^5]\, \sin x = \frac{1}{120}$

live ↗

In$\mathrm{SeriesCoefficient}(\sin(x), \bigl\lbrack x, 0, 5\bigr\rbrack)$Out$\frac{1}{120}$

every coefficient of $\frac{1}{1-x}$ about $0$ is $1$

live ↗

In$\mathrm{SeriesCoefficient}(\frac{1}{1-x}, \bigl\lbrack x, 0, 10\bigr\rbrack)$Out$1$

$[x^4]\, \ln(1+x) = -\frac{1}{4}$

live ↗

In$\mathrm{SeriesCoefficient}(\ln(1+x), \bigl\lbrack x, 0, 4\bigr\rbrack)$Out$-\frac{1}{4}$

Scope #

With a symbolic index $n$ the coefficient is left to the rational function

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In$\mathrm{SeriesCoefficient}(\frac{x}{x-5}, \bigl\lbrack x, 0, n\bigr\rbrack)$Out$\mathrm{SeriesCoefficient}(\frac{x}{x-5}, \bigl\lbrack x, 0, n\bigr\rbrack)$

A rational function with three poles, coefficient left symbolic in $n$

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In$\mathrm{SeriesCoefficient}(\frac{x^3}{(x-3)(x-2)(x-1)}, \bigl\lbrack x, 0, n\bigr\rbrack)$Out$\mathrm{SeriesCoefficient}(\frac{x^3}{(x-3)(x-2)(x-1)}, \bigl\lbrack x, 0, n\bigr\rbrack)$

The log of a reciprocal root is rewritten as $-\frac12\ln(1-5x^2)$

live ↗

In$\mathrm{SeriesCoefficient}(\ln(\frac{1}{\sqrt{1-5x^2}}), \bigl\lbrack x, 0, n\bigr\rbrack)$Out$\mathrm{SeriesCoefficient}(\ln(\frac{1}{\sqrt{1-5x^2}}), \bigl\lbrack x, 0, n\bigr\rbrack)$

live ↗

In$\mathrm{SeriesCoefficient}(\frac{e^{x}}{1-x}, \bigl\lbrack x, 0, n\bigr\rbrack)$Out$\mathrm{SeriesCoefficient}(\frac{e^{x}}{1-x}, \bigl\lbrack x, 0, n\bigr\rbrack)$

live ↗

In$\mathrm{SeriesCoefficient}(\frac{\operatorname{I}_{3}(2x)}{x^7}, \bigl\lbrack x, 0, n\bigr\rbrack)$Out$\mathrm{Piecewise}(\bigl\lbrack\bigl\lbrack\frac{1}{Γ(n/2+3)Γ(n/2+6)}, n\bmod2=0∧-4≤n\bigr\rbrack\bigr\rbrack, 0)$

Properties #

The same, with the sign of the quadratic flipped

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In$\mathrm{SeriesCoefficient}(\ln(\frac{1}{\sqrt{5x^2+1}}), \bigl\lbrack x, 0, n\bigr\rbrack)$Out$\mathrm{SeriesCoefficient}(\ln(\frac{1}{\sqrt{5x^2+1}}), \bigl\lbrack x, 0, n\bigr\rbrack)$

Possible issues #

the residue of $1/x$ at $0$ is a Laurent coefficient, past what the derivative formula covers — declined, not guessed

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In$\mathrm{SeriesCoefficient}(\frac{1}{x}, \bigl\lbrack x, 0, -1\bigr\rbrack)$Out$\mathrm{SeriesCoefficient}(\frac{1}{x}, \bigl\lbrack x, 0, -1\bigr\rbrack)$

$\sin(x)/x$ is singular exactly at the expansion point (removably so) — declined rather than risk the wrong limit

live ↗

In$\mathrm{SeriesCoefficient}(\frac{\sin(x)}{x}, \bigl\lbrack x, 0, 0\bigr\rbrack)$Out$\mathrm{SeriesCoefficient}(\frac{\sin(x)}{x}, \bigl\lbrack x, 0, 0\bigr\rbrack)$

About

Examples	13
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