Asymptotic expansion

For fixed real $\mu \ne 0$, roots of $\tan x=\mu x$ tending to positive infinity approach tangent poles $a_n=(n+1/2)\pi$. Expanding $-\cot (x-a_n)$ givesThe expansion works for either sign of $\mu$ when $|\mu |a_n\gg 1$, but is nonuniform at $\mu =0$, where the roots lie at $n\pi$ instead. The pole labeling need not equal the ordered root index at small $n$.

For positive rates $a,b$, the mean value theorem in the rate gives a bound by $|a-b|x{e}^{-\min (a,b)x}$. Its maximum is $|a-b|/[e\min (a,b)]$. Combining this estimate with coefficient errors bounds a finite spectral sum uniformly on a half-line, including slow scales much longer than one. Positivity of both rates is essential: unstable exponentials need a different time-window or relative-error analysis.

For a formal inverse-power series ${\sum }_{n\ge 0}{a}_n{z}^{-n-1}$, divide its coefficients by $n!$ to form a Borel transform $\sum a_n{t}^n/n!$ and integrate it against $e^{-zt}$. Singularities on the integration ray require a specified contour prescription. Different lateral contours can produce exponentially small differences and the Stokes phenomenon.

If a Borel summation contour meets a simple pole, an upper or lower indentation fixes the exponentially small contribution. For the transform $(1-t^2{)}^{-1}$ on the positive ray, the relevant pole is $t=1$. A lower indentation gives the Cauchy principal value minus $i\pi e^{-z}/2$. Finite geometric expansion gives an exact remainder integral, which enables optimal truncation and error-function smoothing of a Stokes multiplier.

A divergent asymptotic expansion should usually be stopped near its smallest term rather than summed indefinitely. For factorial terms $n!z^{-n}$ the cutoff has $n=|z|+O(1)$. This exposes exponentially small contributions and the Stokes phenomenon. The smallest term alone does not always bound the remainder, especially when a saddle meets a pole.

For a polynomial depending on a small parameter, substitute $t={ϵ}^{p}s$ and compare the exponents of the parameter. Candidate root scales arise when two or more lowest exponents balance. Solve the leading scaled equation and expand about each of its nonzero roots. Different balances can reveal different families of roots; real-root counts must be checked separately.

A polynomial whose degree decreases when its parameter vanishes may have roots escaping to infinity. Such a singular perturbation root is found by balancing powers after rescaling $\lambda ={\epsilon }^{-p}\mu$. If the root enters an exponential, even a lower-order additive root correction can matter on a fixed interval when its leading exponential grows. This phenomenon is different from a repeated root of a fixed polynomial.

If $P({\lambda }_0,0)=0$ and $P_{\lambda }({\lambda }_0,0)\ne 0$, the implicit function theorem gives a local root branch. For analytic coefficients it has an ordinary power series in the perturbation parameter. Its corrections follow by substituting that series into the polynomial and comparing coefficients. A vanishing leading root may require its first nonzero correction to capture a slow exponential mode.

Suppose $a,b,L_0>0$, $L(x)$ is smooth and positive near zero with $L(0)=L_0$, and $d=O(\eta )$ as $\eta \to 0^{+}$. The leading contribution tois obtained by scaling $x=\eta s$ and replacing $L$ by $L_0$. Integration of the resulting reciprocal quadratic gives the displayed formula. For constant $L$ the exact expression isThis proves the half-peak value at $d=0$ and the transition between an interior peak and a pole outside the interval. If $L$ varies, smoothness and positivity on the contributing neighborhood justify its leading replacement; other zeros or singular contributions elsewhere must be excluded.

Exponentially small contributions can change their asymptotic representation when a complex parameter crosses a Stokes direction. An all-orders asymptotic expansion in inverse powers alone does not resolve these terms.

For $z=2n+i\mu \sqrt{n}+O(1)$, the factor $t^{2n}{e}^{z(1-t)}$ has saddle scale $t=1+u/\sqrt{n}$ and becomes $e^{-u^2-i\mu u}$. If the amplitude has a simple pole there, its leading local integral contains $e^{-u^2-i\mu u}/u$. Differentiating its Cauchy principal value with respect to $\mu$ reduces it to a Gaussian integral, producing an error function. The nonsingular amplitude $(1+t{)}^{-1}$ gives $\frac{1}{2}\sqrt{\pi /n}{e}^{-{\mu }^2/4}$.

The pole contour in a Gaussian saddle coalescing with a simple pole changes the multiplier smoothly. For the lower contour, ${\int }_{{\mathcal{C}}_{-}}{e}^{-u^2-i\mu u}du/u=i\pi \mathrm{erfc}(\mu /2)$. This follows by integrating $-i\sqrt{\pi }{e}^{-{\mu }^2/4}$ and adding the indentation contribution $i\pi$. The transition width is inverse square root of the large parameter; on a Stokes line the multiplier takes half its switched-on value.

With the equal-magnitude convention, an anti-Stokes line has $\mathrm{Re}(S_1-S_2)=0$ for competing exponentials. For $e^{\pm x^2/\epsilon }$, its rays are $\arg x=\pi /4+m\pi /2$. Crossing one interchanges which exponential grows. Some literature interchanges the names, so the action-based convention should always be specified.

With the convention used here, a Stokes line is a ray where competing exponential actions have the same imaginary part. A subdominant exponential can acquire a rapidly changing multiplier there. For $e^{\pm x^2/\epsilon }$, these rays have $\arg x=m\pi /2$. The anti-Stokes lines instead have equal exponential magnitudes.

Series reversion constructs the local inverse of a power series. Substituting an ansatz for the inverse and matching successive powers determines its coefficients recursively.

The divide-and-conquer method splits an integral at an intermediate scale, expands each part in the regime where its approximation is uniform, and cancels the artificial splitting parameter after the pieces are recombined.

Functions $({\varphi }_n)$ form an asymptotic sequence at $x_0$ when ${\varphi }_{n+1}=o({\varphi }_n)$. The notationmeans that for every $N\ge 0$,Successive division of the remainder by ${\varphi }_N$ uniquely recovers every coefficient.

For a positive asymptotic sequence $({\varphi }_n)$, define ${\psi }_0={\varphi }_0$ andThen ${\psi }_n\sim {\varphi }_n$ term by term and $({\psi }_n)$ is again an asymptotic sequence.

For ${\chi }_0={\varphi }_0$ and ${\chi }_n=\sqrt{{\varphi }_{n-1}{\varphi }_n}$, the sequence $({\chi }_n)$ is asymptotic, but ${\chi }_n/{\varphi }_n\to \infty$ for every $n\ge 1$.

Even if ${\varphi }_n\sim {\psi }_n$ for every $n$, an expansion in the scale $({\varphi }_n)$ need not induce one in $({\psi }_n)$. A perturbation of ${\psi }_0$ that is smaller than ${\varphi }_0$ but larger than ${\varphi }_1$ supplies a counterexample.

If a smooth real phase $\varphi$ has a unique nondegenerate interior maximum at $t_0$, thenFor a maximum at an endpoint $b$ with ${\varphi }^{\prime }(b)>0$, the leading factor is $a(b)e^{x\varphi (b)}/(x{\varphi }^{\prime }(b))$.

If a real smooth phase has its strict maximum at the upper endpoint $b$ and $\varphi \prime (b)>0$, then ${\int }_a^{b}a(t)e^{\lambda \varphi (t)}dt\sim a(b)e^{\lambda \varphi (b)}/(\lambda \varphi \prime (b))$. The endpoint coordinate $b-t$ has scale $1/\lambda$. At a nondegenerate stationary endpoint the scale is instead ${\lambda }^{-1/2}$ and the leading contribution is half a Gaussian integral.

A Laplace integral contains a large parameter in an exponential weight, typically in the form $\int a(t)e^{x\varphi (t)}dt$ or ${\int }_0^b{e}^{-xt}f(t)dt$. Its asymptotic behavior is controlled by the points where the real part of the phase is largest.

A linear differential-equation solution may be sought as ${\int }_C{e}^{xt}f(t)dt$. Integration by parts transfers the factor $x$ to derivatives of $f$, giving an equation for the amplitude and boundary terms. Admissible contours must cancel those terms and make the integral converge.

If the maximum $t_0$ of the smooth real phase $\varphi$ is an endpoint, ${\varphi }^{\prime }(t_0)=0$, and ${\varphi }^{\prime \prime }(t_0)<0$, the local Gaussian is integrated over only one side of its peak. Consequentlywith the analogous formula for a right endpoint.

If a Laplace integral has finitely many separated nondegenerate global maxima with the same phase value, its leading term is the sum of their local Gaussian contributions. An interior maximum contributes a full Gaussian integral, while a stationary endpoint maximum contributes half of one.

Suppose a real phase has its unique minimum at the left endpoint and $\varphi (t)-\varphi (0)\sim c{t}^m$, with $c>0$ and integer $m\ge 1$. For a continuous amplitude $a$ with $a(0)\ne 0$, localization followed by $s=(\nu c{)}^{1/m}t$ givesThis uses the Gamma integral. It assumes sufficient decay away from the minimum to make the remaining integral negligible. A cubic endpoint has width ${\nu }^{-1/3}$; an ordinary quadratic endpoint has width ${\nu }^{-1/2}$.

For a Laplace integral with local phase $\varphi (t;\delta )=\delta t+b{t}^3/3+\cdots$, $b>0$, an interior minimum approaches the endpoint as $\delta \uparrow 0$. The distinguished limit is $t=O({\nu }^{-1/3})$, $\delta =O({\nu }^{-2/3})$. With $s=(\nu b{)}^{1/3}t$, its leading local integral isThe cubic Laplace transition integral joins the ordinary endpoint estimate for positive $\delta$, the cubic endpoint estimate at zero, and the interior Laplace's method estimate for negative $\delta$. Separate fixed-parameter formulas fail to be uniform when the minimum is within its own width of the endpoint.

The discrete Laplace method estimates a sum whose positive terms concentrate near the maximum of a smooth exponent. When the Gaussian concentration width contains asymptotically many lattice points, the sum is a Riemann-sum version of Laplace's method with the corresponding lattice-spacing factor.

After $t=zs$, the phase in the integral for ${\Gamma }^{\prime }(z)$ becomes $\log s-s$, with maximum $-1$ and second derivative $-1$ at $s=1$. Hence

A contour integral with a large exponential parameter is deformed through saddle points along curves on which the real part of the phase decreases most rapidly and the imaginary part is locally constant.

For an integrand $e^{kf(z)}$, a contour along which $\mathrm{Im}f$ is constant and $\mathrm{Re}f$ decreases away from the contributing endpoint or saddle. This suppresses oscillation while localizing the large-parameter integral.

For $e^{-\lambda z^2}$, the saddle point descent is the real axis and ascent is the imaginary axis. A descent path from a nonzero endpoint $b$ has $z(s{)}^2=b^2+s$, $s\ge 0$, with the continuous square-root branch from $b$. It ends at positive or negative real infinity according to the sign of $\mathrm{Re}b$. The oriented outward endpoint contribution in steepest descent is $e^{-\lambda b^2}/(2\lambda b)$.

For a Gaussian integral from $-1/\sqrt{2}$ to $e^{i\theta }$, a saddle point contribution dominates for $0\le \theta \le \pi /4$. The moving endpoint contribution in steepest descent dominates for $\pi /4<\theta <5\pi /6$. At $5\pi /6$ the two endpoints have equal exponential rates and both are needed; for larger angles the fixed endpoint dominates. Saddle connectivity switches at $\pi /2$, but that change is subdominant in the leading endpoint regime.

For a holomorphic phase $\varphi$, a saddle point is a critical point $z_0$ with ${\varphi }^{\prime }(z_0)=0$. At a simple saddle, ${\varphi }^{\prime \prime }(z_0)\ne 0$, and the quadratic Taylor term determines the local descent directions.

A saddle-point approximation expands an exponent around a stationary configuration and integrates its leading Gaussian fluctuations. A large parameter multiplying the exponent suppresses higher fluctuations.

When two quadratic saddles coalesce and the first nonzero phase derivative is cubic, the contributing neighborhood has width $O({\lambda }^{-1/3})$ rather than $O({\lambda }^{-1/2})$. The leading contour integral reduces to a Gamma-function or Airy-function constant.

When a simple pole approaches a saddle within its Gaussian width, separate residue and saddle approximations cease to be uniform. A local change of variable reduces the transition to a complementary error function whose multiplier smoothly switches the pole residue across the deformed contour.

Let $G^{\prime }(z_0)=0$, $G^{\prime \prime }(z_0)\ne 0$, and let $F,G$ be analytic. Assume a contributing contour can be deformed onto two descent arms of the method of steepest descent with one clockwise semicircular avoidance of the pole at $z_0$, and that its other portions are negligible. For the integrand $F(z)e^{-i\lambda G(z)}/(z-z_0)$, a local coordinate with $G-G_0=-i{w}^2/2$ givesThe odd pole term has zero Cauchy principal value on the real arms and integral $-i\pi F_0$ on the clockwise indentation. The constant regular term integrates to $\sqrt{2\pi /\lambda }$ times its coefficient, proving the expansion and generic correction order. A degenerate saddle has a different local width and possibly a different indentation angle; analyticity and a single critical point alone do not imply this result. If $F_0=0$, retain the additive expansion rather than writing asymptotic equivalence to zero.

For a simple saddle point, a nearby pole coalesces with its contributing region when their separation is of order the inverse square root of the large parameter. A local quadratic phase coordinate changes the singular part to a Gaussian pole integral. Adding any residue crossed by the contour deformation gives a Faddeeva function rather than two separately singular approximations. The regular part of the transformed amplitude remains smaller by the ordinary saddle width.

If ${\varphi }^{\prime }(z_0)=0$, ${\varphi }^{\prime \prime }(z_0)\ne 0$, and the oriented steepest-descent tangent has angle $\alpha$, thenThe tangent condition is

If a deformed contour starts at an endpoint $z_0$ with ${\varphi }^{\prime }(z_0)\ne 0$ and initially enters a decaying sector, local integration gives the leading contribution $-a(z_0)e^{\lambda \varphi (z_0)}/[\lambda {\varphi }^{\prime }(z_0)]$, with the sign adjusted to the contour orientation.

An endpoint at infinity contributes when the region controlling an asymptotic integral recedes without bound as the large parameter changes. Expanding the phase and amplitude at infinity reveals the rescaling that converts the distant tail into a finite limiting integral.