Asymptotic expansion

Definitions

If ${\displaystyle {\phi }_n}$ is a sequence of functions of asymptotic variable ${\displaystyle x}$ on some domain, and if ${\displaystyle L}$ is a limit point of the domain, then the sequence is an asymptotic scale if for every ${\displaystyle n}$

${\displaystyle {\phi }_{n+1}(x)=o({\phi }_n(x))(x\to L)}$.

Here, ${\displaystyle o}$ is the little-o notation. Each function within the scale is called a gauge function. Equivalently, each successive gauge function in the scale vanishes or grows strictly slower than its predecessor as ${\displaystyle x\to L}$. Commonly, ${\displaystyle L}$ is taken to be zero or infinity. If ${\displaystyle L}$ is a finite number, it is called a distinguished point of the scale.

If ${\displaystyle f}$ is a function on the domain of the asymptotic scale, then ${\displaystyle f}$ has a standard asymptotic expansion with respect to the scale as a formal series:

${\displaystyle \sum _{m=0}^n{a}_m{\phi }_m(x)}$

if

${\displaystyle f(x)-\sum _{m=0}^{n-1}{a}_m{\phi }_m(x)=O({\phi }_n(x))(x\to L)}$

or the weaker condition

${\displaystyle f(x)-\sum _{m=0}^{n-1}{a}_m{\phi }_m(x)=o({\phi }_{n-1}(x))(x\to L)}$

is satisfied. Here, ${\displaystyle O}$ is the Big-O notation. If one or the other holds for all ${\displaystyle n}$, this is indicated as:

${\displaystyle f(x)\sim \sum _{m=0}^{\mathrm{\infty }}{a}_m{\phi }_m(x)(x\to L).}$

A convergent series for any fixed ${\displaystyle x}$ approximates the function as the number of terms approaches infinity. In contrast, the asymptotic series for a fixed number of terms approximates the function as ${\displaystyle x}$ approaches the limit ${\displaystyle L}$.

The function ${\displaystyle f}$ is a sum of its truncated asymptotic series, identified by its truncation index ${\displaystyle n}$, and a remainder, ${\displaystyle r_n(x)}$:

${\displaystyle f(x)=\sum _{m=0}^{n-1}{a}_m{\phi }_m(x)+r_n(x).}$

This finite series approximates the function with a truncation error equal to the remainder. The series tail, ${\displaystyle t_n(x)}$, is the remaining part of the series:

${\displaystyle t_n(x)=\sum _{m=n}^{\mathrm{\infty }}{a}_m{\phi }_m(x).}$

Regularization of an asymptotic series identifies the underlying function, ${\displaystyle f(x)}$, that generates the asymptotic series expansion, assigning the series a unique value for each point in its domain.

Properties

Unique expansion

The asymptotic expansion of a function ${\displaystyle f(x)}$ is unique for a specific region, scale and limit. The function has uniquely determined coefficients ${\displaystyle \{a_n\}}$ in the following way: $${\displaystyle \begin{array}{cc}{a}_0& =\underset{x\to L}{lim}\frac{f(x)}{{\phi }_0(x)}\\ a_1& =\underset{x\to L}{lim}\frac{f(x)-a_0{\phi }_0(x)}{{\phi }_1(x)}\\ & ⋮\\ a_n& =\underset{x\to L}{lim}\frac{f(x)-\sum _{m=0}^{n-1}{a}_m{\phi }_m(x)}{{\phi }_n(x)}\end{array}}$$ where ${\displaystyle L}$ is the limit point of this asymptotic expansion (may be ${\displaystyle \pm \mathrm{\infty }}$).

However, a function may have distinct expansions using distinct asymptotic scales. An example is: $${\displaystyle \begin{array}{cc}{\displaystyle \frac{1}{(1+x)}}& \sim \sum _{m=1}^{\mathrm{\infty }}(-1{)}^m{x}^{-m}(x\to \mathrm{\infty })\\ {\displaystyle \frac{1}{(1+x)}}& \sim \sum _{m=1}^{\mathrm{\infty }}(x-1)x^{-2m}(x\to \mathrm{\infty })\end{array}}$$

Sector of validity

In complex analysis, the limit of a function as it approaches a point in the complex plane may depend on the path's direction. Therefore, it is necessary to specify the sector of validity, a wedge-shaped region where the function has a specified limit. The sector of validity's angular width is called its opening angle and its boundaries are rays whose angles are given by the ${\displaystyle \arg (x)}$ function. For complex-variable functions, a complete description of a divergent asymptotic series must specify its sector of validity, as this region varies by series.

Nonunique function

If the function ${\displaystyle f(x)}$ has this power series expansion:

${\displaystyle f(x)\sim \sum _{m=0}^{\mathrm{\infty }}{a}_m{x}^m(x\to 0),}$

adding an exponential term, ${\displaystyle e^{-\frac{1}{x^2}}}$, leads to the same expansion as ${\displaystyle x\to 0}$ along the real axis:

${\displaystyle f(x)+e^{-\frac{1}{x^2}}\sim \sum _{m=0}^{\mathrm{\infty }}{a}_m{x}^m(x\to 0).}$

This occurs because the exponential function's own Taylor coefficients at the origin all vanish. Thus, two distinct functions can share the exact same asymptotic expansion.

Subdominance

If two functions are equivalent ${\displaystyle f(x)\sim g(x)}$ as ${\displaystyle x\to x_0}$ within a sector of validity, their difference ${\displaystyle \delta (x)=f(x)-g(x)}$ is subdominant to both functions. A function subdominant in one sector may become dominant in a neighboring sector, a process associated with the Stokes phenomenon..

Examples

• Gamma function (Stirling's approximation):

${\displaystyle \frac{e^x}{x^x\sqrt{2\pi x}}\mathrm{\Gamma }(x+1)\sim 1+\frac{1}{12x}+\frac{1}{288x^2}-\frac{139}{51840x^3}-\cdots ,(x\to \mathrm{\infty }),\text{for }|\arg (x)|<\pi .}$

• Exponential integral:

${\displaystyle x{e}^x{E}_1(x)\sim \sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^{n}n!}{x^n},(x\to \mathrm{\infty }),\text{for }|\arg (x)|<\pi .}$

• Logarithmic integral for real variable ${\displaystyle x}$:

${\displaystyle \mathrm{li}(x)\sim \frac{x}{\ln x}\sum _{k=0}^{\mathrm{\infty }}\frac{k!}{(\ln x{)}^k},(x\to \mathrm{\infty }).}$

• Riemann zeta function:

${\displaystyle \zeta (s)\sim \sum _{n=1}^N{n}^{-s}+\frac{N^{1-s}}{s-1}-\frac{N^{-s}}{2}+N^{-s}\sum _{m=1}^{\mathrm{\infty }}\frac{B_{2m}{s}^{\overline{2m-1}}}{(2m)!N^{2m-1}},\text{for }|\arg (s)|<\pi ,\mathrm{Re}(s)>-2n,n=1,2,3,\dots }$

where ${\displaystyle B_{2m}}$ are Bernoulli numbers and ${\displaystyle s^{\overline{2m-1}}}$ is a rising factorial. This expansion is valid for all complex ${\displaystyle s}$ and is often used to compute the zeta function by using a large enough value of ${\displaystyle N}$, for instance ${\displaystyle N>|s|.}$

• Complementary error function:

${\displaystyle \sqrt{\pi }x{e}^{x^2}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}(x)\sim 1+\sum _{n=1}^{\mathrm{\infty }}(-1{)}^n\frac{(2n-1)!!}{(2x^2{)}^n}(x\to \mathrm{\infty }),\text{for }|\arg (x)|<\frac{3\pi }{4}}$

where ${\displaystyle (2n-1)!!}$ is the double factorial.

Worked example

This example generates an asymptotic expansion for a function related to the exponential integral defined using the Cauchy principal value (PV):

${\displaystyle e^{-1/t}\mathrm{Ei}\left(\frac{1}{t}\right)=-e^{-1/t}\mathrm{PV}{\int }_{-\frac{1}{t}}^{\mathrm{\infty }}\frac{e^{-u}}{u}du.}$

The substitution ${\displaystyle w=ut+1}$ transforms this expression into:

${\displaystyle e^{-1/t}\mathrm{Ei}\left(\frac{1}{t}\right)=\mathrm{PV}{\int }_0^{\mathrm{\infty }}\frac{e^{-w/t}}{1-w}dw.}$

Applying the finite geometric series formula to the integrand's denominator expresses the function as a sum of integrals:

${\displaystyle e^{-1/t}\mathrm{Ei}\left(\frac{1}{t}\right)=\sum _{m=0}^{n-1}{\int }_0^{\mathrm{\infty }}{w}^m{e}^{-w/t}dw+\mathrm{PV}{\int }_0^{\mathrm{\infty }}\frac{w^n{e}^{-w/t}}{1-w}dw.}$

Evaluating these integrals generates a finite series with factorial coefficients and an integral remainder:

${\displaystyle e^{-1/t}\mathrm{Ei}\left(\frac{1}{t}\right)=\sum _{m=0}^{n-1}m!t^{m+1}+\mathrm{PV}{\int }_0^{\mathrm{\infty }}\frac{w^n{e}^{-w/t}}{1-w}dw.}$

Although the series is divergent, truncating the series may provide a good approximation of the exponential integral for sufficiently small ${\displaystyle t}$.

The full asymptotic expansion includes all series terms:

${\displaystyle e^{-1/t}\mathrm{Ei}\left(\frac{1}{t}\right)\sim \sum _{m=0}^{\mathrm{\infty }}m!t^{m+1}(t\to 0^{+}).}$

Repeated integration by parts is an alternative method to derive this identical expansion. Substituting ${\displaystyle x=-\frac{1}{t}}$ and noting that ${\displaystyle \mathrm{Ei}(x)=-E_1(-x)}$ recovers the expansion given earlier in the article:

${\displaystyle \mathrm{Ei}(x)=\frac{e^x}{x}\left(\sum _{m=0}^n\frac{m!}{x^m}+e_n(x)\right),e_n(x)\equiv (n+1)!x{e}^{-x}{\int }_{-\mathrm{\infty }}^x\frac{e^t}{t^{n+2}}dt.}$

For any fixed ${\displaystyle x}$, the absolute value of the error term ${\displaystyle |e_n(x)|}$ decreases, then increases as ${\displaystyle n}$ increases. The minimum occurs at ${\displaystyle n\sim |x|}$, at which point ${\displaystyle |e_n(x)|\le \sqrt{\frac{2\pi }{|x|}}{e}^{-|x|}}$. Exponentially small truncation errors are common for optimally truncated series.

Asymptotic expansions may arise when a series is integrated beyond its boundary of convergence.

Regularization methods

Optimal truncation

These regularization methods assign a value to an asymptotic series using truncated series.

The superasymptotic expansion is the truncated series which minimizes the magnitude of the remainder term. A remainder formula or remainder upper bound formula may determine the optimal truncation index ${\displaystyle n}$. If an exact formula is unavailable, heuristic methods may determine the optimum truncation index such as truncating the series immediately before or at its minimum term (summation to the least term), or using formulas that determine the optimal index ${\displaystyle n}$ from the value of the asymptotic variable ${\displaystyle x}$.

A term ${\displaystyle \delta (x)}$ is said to lie beyond-all-orders of an asymptotic expansion if the term is asymptotically smaller than every gauge function ${\displaystyle {\phi }_n(x)}$ of the expansion:

${\displaystyle \delta (x)=o({\phi }_n(x))(x\to L).}$

Commonly, an optimally truncated power series expansion at the origin has a beyond-all-orders truncation error of order ${\displaystyle O(e^{-q/x})}$ with positive constant ${\displaystyle q}$ as ${\displaystyle x\to 0}$ in the sector ${\displaystyle |\arg (x)|<\pi /2}$. At the origin, this truncation error is non-analytic and cannot be expressed as a Taylor series. The term asymptotics beyond-all-orders refers to the collection of methods used to measure the effects of these beyond all order terms.

For an alternating series, the remainder is often approximately half of the series' least term. Adding half of the least term to the sum of the preceding terms often generates a highly accurate approximation.

The hyperasymptotic expansion replaces the remainder of a superasymptotic expansion with an optimally truncated second series and second smaller remainder, often using a different scale. Repeating this process generates exponentially smaller remainder terms.

Summation of the series tail

These regularization methods compute a series remainder by applying Borel summation or Mellin-Barnes regularization to the series tail. The expansion's underlying function is the sum of this computed remainder and the truncated series.

Borel summation

When a truncated series has a factorially divergent tail, Borel summation evaluates the remainder as the product of the tail's first term and an integral called a terminant.

For an alternating-sign tail:

${\displaystyle \sum _{m=n}^{\mathrm{\infty }}\mathrm{\Gamma }(m+\alpha +1)(-x{)}^m,0\le \alpha <1,|\arg (x)|<\pi ,}$

Borel summation generates a type I terminant ${\displaystyle {\mathrm{\Lambda }}_s(x)}$, defined by a Cauchy principal value integral:

${\displaystyle {\mathrm{\Lambda }}_s(x)=\frac{1}{\mathrm{\Gamma }(s+1)}\mathrm{PV}{\int }_0^{\mathrm{\infty }}\frac{t^s{e}^{-t}}{1+xt}dt.}$

Multiplying this terminant by the tail's first term generates the exact remainder:

${\displaystyle \mathrm{\Gamma }(n+\alpha +1)(-x{)}^n{\mathrm{\Lambda }}_{n+\alpha }(x).}$

For a same-sign tail:

${\displaystyle \sum _{m=n}^{\mathrm{\infty }}\mathrm{\Gamma }(m+\alpha +1)x^m,0\le \alpha <1,}$

the positive real axis forms a Stokes line. Borel summation generates a type II terminant ${\displaystyle {\overline{\mathrm{\Lambda }}}_s(-x)}$ with the Stokes constant ${\displaystyle S}$:

${\displaystyle \begin{array}{cc}& {\overline{\mathrm{\Lambda }}}_s(-x)={\mathrm{\Lambda }}_s(-x)+2S{\displaystyle \frac{i\pi x^{-s-1}{e}^{-1/x}}{\mathrm{\Gamma }(s+1)}},\\ & S=\{\begin{array}{ll}+{\displaystyle \frac{1}{2}},& \text{if }0<\arg (x)<2\pi ,\\ 0,& \text{if }\arg (x)=0,\\ -{\displaystyle \frac{1}{2}},& \text{if }-2\pi <\arg (x)<0.\end{array}\end{array}}$

The resulting remainder is the product:

${\displaystyle \mathrm{\Gamma }(n+\alpha +1)x^n{\overline{\mathrm{\Lambda }}}_{n+\alpha }(-x).}$

For more general divergent tails, a linear combination of late-series terms and terminants, called a distribution over basic terminants, represents the remainder.

Mellin–Barnes tail regularization

Mellin–Barnes regularization evaluates the series tail separately for alternating-sign coefficients and same-sign coefficients.

For an alternating-sign series tail:

${\displaystyle \sum _{m=n}^{\mathrm{\infty }}f(m)(-x^{\beta }{)}^m,}$

Mellin–Barnes regularization computes this remainder:

${\displaystyle {\int }_{c-i\mathrm{\infty }}^{c+i\mathrm{\infty }}\frac{x^{\beta s}f(s)}{e^{-i\pi s}-e^{i\pi s}}ds,\frac{-\pi -{\epsilon }_1}{\beta }<\arg (x)<\frac{\pi +{\epsilon }_2}{\beta },}$

where ${\displaystyle n-1<c=\mathrm{Re}(s)<n}$. The asymptotic behavior of the integrand as ${\displaystyle s}$ approaches the limits of integration determines the constants ${\displaystyle {\epsilon }_1}$ and ${\displaystyle {\epsilon }_2.}$

For a same-sign series tail:

${\displaystyle \sum _{m=n}^{\mathrm{\infty }}f(m)(x^{\beta }{)}^m,}$

the positive real axis forms a Stokes line. For the sector above the Stokes line, the method generates this remainder:

${\displaystyle {\int }_{c-i\mathrm{\infty }}^{c+i\mathrm{\infty }}\frac{x^{\beta s}{e}^{i\pi s}f(s)}{e^{-i\pi s}-e^{i\pi s}}ds+\frac{1}{2}{\int }_{c-i\mathrm{\infty }}^{c+i\mathrm{\infty }}{x}^{\beta s}f(s)ds,0<\arg (x)<\frac{2\pi -B}{\beta },}$

where ${\displaystyle n-1<c=\mathrm{Re}(s)<n.}$

For the sector below the Stokes line, the remainder becomes:

${\displaystyle {\int }_{c-i\mathrm{\infty }}^{c+i\mathrm{\infty }}\frac{x^{\beta s}{e}^{-i\pi s}f(s)}{e^{-i\pi s}-e^{i\pi s}}ds-\frac{1}{2}{\int }_{c-i\mathrm{\infty }}^{c+i\mathrm{\infty }}{x}^{\beta s}f(s)ds,\frac{-2\pi +A}{\beta }<\arg (x)<0,}$

where ${\displaystyle n-1<c=\mathrm{Re}(s)<n.}$ The asymptotic behavior of the integrand as ${\displaystyle s}$ approaches the limits of integration determines the constants ${\displaystyle A}$ and ${\displaystyle B.}$ Additional integrals, convergent in overlapping sectors, allow analytic continuation in the complex plane for both types of series.

Summation of the full series

This regularization method applies Borel summation or Mellin-Barnes regularization to the entire series. Computing the Borel sum of a typical Gevrey order 1 asymptotic series requires a Borel transform, analytic continuation, and a Laplace transform.

The Borel transform converts this divergent power series with zero radius of convergence into a convergent series with a nonzero radius of convergence called the Borel transformed series. A series acceleration method can accurately sum a slow converging series.

The analytic continuation of the Borel transformed series is the Borel transformed function. The Borel transformed function's domain now extends across the complex plane, excluding the function's singularities. Comparing the transformed series coefficients to established tables of function expansions and coefficient sequences may identify the analytic continuation as a special function, an integral representation, or a generating function. An alternative approach uses rational function approximation of the transformed series, guided by singularity analysis and conformal mapping. A bijective conformal map (uniform map) is especially effective for accurate approximation. In comparison to the Padé approximant, the Adaptive Antoulas–Anderson (AAA) algorithm is numerically more stable and less prone to generating spurious poles.

Finally, the Laplace transform converts the Borel transformed function to the Borel sum, a finite value assigned to the original divergent series.

Convergent alternative expansions

Series such as inverse factorial series (or Nörlund series), binary rational inverse factorial series, or modified Hadamard series are alternatives to asymptotic expansions because they may converge for functions with divergent asymptotic series.

The inverse factorial series (or Nörlund series) with coefficients ${\displaystyle c_m}$ has the following form: $${\displaystyle f(x)=c_0+\sum _{m=1}^{\mathrm{\infty }}\frac{c_m}{x(x+1)(x+2)\cdots (x+m-1)}.}$$

A binary rational inverse factorial series uses a double summation over the coefficients ${\displaystyle c_{k,m}}$; it has the following form: $${\displaystyle f(z)=\sum _{k=1}^{\mathrm{\infty }}\sum _{m=0}^{\mathrm{\infty }}\frac{c_{k,m}}{2^{km}}\frac{m!}{(2^{k}z+1{)}^{m+1}}.}$$ This binary rational variant typically converges more rapidly than the standard inverse factorial series.

A modified Hadamard series is a convergent series designed to overcome the divergence or slow convergence of traditional asymptotic expansions. Each term in the series has a normalized incomplete gamma function that can gradually decrease the magnitude of later terms. Selecting optimal integration intervals and rearranging how later terms are summed leads to high accuracy.