method of steepest descent

The method of steepest descent is a method to approximate a complex integral of the form$$I(\backslash lambda)\; =\; \backslash int\_Cf(z)e^\{\backslash lambda\; g(z)\}\backslash ,\backslash mathrm\{d\}z$$for large $\backslash lambda\; \backslash rightarrow\; \backslash infty$, where $f(z)$ and $g(z)$ are analytic functions of $z$. Because the integrand is analytic, the contour $C$ can be deformed into a new contour $C\text{'}$ without changing the integral. In particular, one seeks a new contour on which the imaginary part, denoted $\backslash Im\; (\backslash cdot)$, of $g(z)\; =\; \backslash Re\; [g(z)]\; +\; i\; \backslash ,\; \backslash Im[g(z)]$ is constant ($\backslash Re\; (\backslash cdot)$ denotes the real part). Then $$I(\backslash lambda)\; =\; e^\{i\; \backslash lambda\; \backslash Im\backslash \{g(z)\backslash \}\}\; \backslash int\_\{C\text{'}\}f(z)e^\{\backslash lambda\; \backslash Re\; \backslash \{g(z)\backslash \}\}\backslash ,\backslash mathrm\{d\}z,$$and the remaining integral can be approximated with other methods like Laplace's method.{{Cite book|last1=Bender|first1=Carl M.|url=http://link.springer.com/10.1007/978-1-4757-3069-2|title=Advanced Mathematical Methods for Scientists and Engineers I|last2=Orszag|first2=Steven A.|date=1999|publisher=Springer New York|isbn=978-1-4419-3187-0|location=New York, NY|language=en|doi=10.1007/978-1-4757-3069-2}}

The method is called the method of steepest descent because for analytic $g(z)$, constant phase contours are equivalent to steepest descent contours.

If $g(z)\; =\; X(z)\; +\; i\; Y(z)$ is an analytic function of $z\; =\; x\; +\; i\; y$, it satisfies the Cauchy–Riemann equations$$\backslash frac\{\backslash partial\; X\}\{\backslash partial\; x\}$$

=

\frac{\partial Y}{\partial y}

\qquad \text{and} \qquad

\frac{\partial X}{\partial y}

=

- \frac{\partial Y}{\partial x}

.Then $$\backslash frac\{\backslash partial\; X\}\{\backslash partial\; x\}\; \backslash frac\{\backslash partial\; Y\}\{\backslash partial\; x\}\; +\; \backslash frac\{\backslash partial\; X\}\{\backslash partial\; y\}\; \backslash frac\{\backslash partial\; Y\}\{\backslash partial\; y\}\; =\; \backslash nabla\; X\; \backslash cdot\; \backslash nabla\; Y\; =\; 0,$$so contours of constant phase are also contours of steepest descent.

Let {{math| f, S : Cⁿ → C}} and {{math|C ⊂ Cⁿ}}. If

:

where $\backslash Re\; (\backslash cdot)$ denotes the real part, and there exists a positive real number {{math|λ₀}} such that

:

then the following estimate holds:A modified version of Lemma 2.1.1 on page 56 in {{harvtxt|Fedoryuk|1987}}.

:$\backslash left|\; \backslash int\_\{C\}\; f(x)\; e^\{\backslash lambda\; S(x)\}\; dx\; \backslash right|\; \backslash leqslant\; \backslash text\{const\}\backslash cdot\; e^\{\backslash lambda\; M\},\; \backslash qquad\; \backslash forall\; \backslash lambda\; \backslash in\; \backslash mathbb\{R\},\; \backslash quad\; \backslash lambda\; \backslash geqslant\; \backslash lambda\_0.$

Proof of the simple estimate:

:$\backslash begin\{align\}$

\left| \int_{C} f(x) e^{\lambda S(x)} dx \right| &\leqslant \int_C |f(x)| \left|e^{\lambda S(x)} \right| dx \\

&\equiv \int_{C} |f(x)| e^{\lambda M} \left | e^{\lambda_0 (S(x)-M)} e^{(\lambda-\lambda_0)(S(x)-M)} \right| dx \\

&\leqslant \int_C |f(x)| e^{\lambda M} \left| e^{\lambda_0 (S(x)-M)} \right| dx && \left| e^{(\lambda-\lambda_0)(S(x) - M)} \right| \leqslant 1 \\

&= \underbrace{e^{-\lambda_0 M} \int_{C} \left| f(x) e^{\lambda_0 S(x)} \right| dx}_{\text{const}} \cdot e^{\lambda M}.

\end{align}

Let {{mvar|x}} be a complex {{mvar|n}}-dimensional vector, and

:$S\text{'}\text{'}\_\{xx\}(x)\; \backslash equiv\; \backslash left(\; \backslash frac\{\backslash partial^2\; S(x)\}\{\backslash partial\; x\_i\; \backslash partial\; x\_j\}\; \backslash right),\; \backslash qquad\; 1\backslash leqslant\; i,\backslash ,\; j\backslash leqslant\; n,$

denote the Hessian matrix for a function {{math|S(x)}}. If

:$\backslash boldsymbol\{\backslash varphi\}(x)\; =\; (\backslash varphi\_1(x),\; \backslash varphi\_2(x),\; \backslash ldots,\; \backslash varphi\_k(x))$

is a vector function, then its Jacobian matrix is defined as

:$\backslash boldsymbol\{\backslash varphi\}\_x\text{'}\; (x)\; \backslash equiv\; \backslash left(\; \backslash frac\{\backslash partial\; \backslash varphi\_i\; (x)\}\{\backslash partial\; x\_j\}\; \backslash right),\; \backslash qquad\; 1\; \backslash leqslant\; i\; \backslash leqslant\; k,\; \backslash quad\; 1\; \backslash leqslant\; j\; \backslash leqslant\; n.$

A non-degenerate saddle point, {{math|z⁰ ∈ Cⁿ}}, of a holomorphic function {{math|S(z)}} is a critical point of the function (i.e., {{math|∇S(z⁰) {{=}} 0}}) where the function's Hessian matrix has a non-vanishing determinant (i.e., $\backslash det\; S\text{'}\text{'}\_\{zz\}(z^0)\; \backslash neq\; 0$).

The following is the main tool for constructing the asymptotics of integrals in the case of a non-degenerate saddle point:

The Morse lemma for real-valued functions generalizes as followsLemma 3.3.2 on page 113 in {{harvtxt|Fedoryuk|1987}} for holomorphic functions: near a non-degenerate saddle point {{math|z⁰}} of a holomorphic function {{math|S(z)}}, there exist coordinates in terms of which {{math|S(z) − S(z⁰)}} is exactly quadratic. To make this precise, let {{mvar|S}} be a holomorphic function with domain {{math|W ⊂ Cⁿ}}, and let {{math|z⁰}} in {{mvar|W}} be a non-degenerate saddle point of {{mvar|S}}, that is, {{math|∇S(z⁰) {{=}} 0}} and $\backslash det\; S$_{zz}(z^0) \neq 0. Then there exist neighborhoods {{math|U ⊂ W}} of {{math|z⁰}} and {{math|V ⊂ Cn}} of {{math|w {{=}} 0}}, and a bijective holomorphic function {{math|φ : V → U}} with {{math|φ(0) {{=}} z''⁰}} such that

:$\backslash forall\; w\; \backslash in\; V:\; \backslash qquad\; S(\backslash boldsymbol\{\backslash varphi\}(w))\; =\; S(z^0)\; +\; \backslash frac\{1\}\{2\}\; \backslash sum\_\{j=1\}^n\; \backslash mu\_j\; w\_j^2,\; \backslash quad\; \backslash det\backslash boldsymbol\{\backslash varphi\}\_w\text{'}(0)\; =\; 1,$

Here, the {{math|μ_j}} are the eigenvalues of the matrix $S\_\{zz\}\text{'}\text{'}(z^0)$.

File:Complex Morse Lemma Illustration.pdf

{{math proof|title=Proof of complex Morse lemma|proof=

The following proof is a straightforward generalization of the proof of the real Morse Lemma, which can be found in.{{harvtxt|Poston|Stewart|1978}}, page 54; see also the comment on page 479 in {{harvtxt|Wong|1989}}.

We begin by demonstrating

:Auxiliary statement. Let {{math| f  : Cⁿ → C}} be holomorphic in a neighborhood of the origin and {{math| f (0) {{=}} 0}}. Then in some neighborhood, there exist functions {{math|g_i : Cⁿ → C}} such that $$f(z)\; =\; \backslash sum\_\{i=1\}^n\; z\_i\; g\_i(z),$$ where each {{math|g_i}} is holomorphic and $$g\_i(0)\; =\; \backslash left.\; \backslash tfrac\{\backslash partial\; f(z)\}\{\backslash partial\; z\_i\}\; \backslash right|\_\{z=0\}.$$

From the identity

:$f(z)\; =\; \backslash int\_0^1\; \backslash frac\{d\}\{dt\}\; f\; \backslash left\; (t\; z\_1,\backslash cdots,\; t\; z\_n\; \backslash right\; )\; dt\; =\; \backslash sum\_\{i=1\}^n\; z\_i\; \backslash int\_0^1\; \backslash left.\; \backslash frac\{\backslash partial\; f(z)\}\{\backslash partial\; z\_i\}\backslash right|\_\{z=(t\; z\_1,\; \backslash ldots,\; t\; z\_n)\}\; dt,$

we conclude that

:$g\_i(z)\; =\; \backslash int\_0^1\; \backslash left.\; \backslash frac\{\backslash partial\; f(z)\}\{\backslash partial\; z\_i\}\backslash right|\_\{z=(t\; z\_1,\; \backslash ldots,\; t\; z\_n)\}\; dt$

and

:$g\_i(0)\; =\; \backslash left.\; \backslash frac\{\backslash partial\; f(z)\}\{\backslash partial\; z\_i\}\; \backslash right|\_\{z=0\}.$

Without loss of generality, we translate the origin to {{math|z⁰}}, such that {{math|z⁰ {{=}} 0}} and {{math|S(0) {{=}} 0}}. Using the Auxiliary Statement, we have

:$S(z)\; =\; \backslash sum\_\{i=1\}^n\; z\_i\; g\_i\; (z).$

Since the origin is a saddle point,

:$\backslash left.\; \backslash frac\{\backslash partial\; S(z)\}\{\backslash partial\; z\_i\}\; \backslash right|\_\{z=0\}\; =\; g\_i(0)\; =\; 0,$

we can also apply the Auxiliary Statement to the functions {{math|g_i(z)}} and obtain

{{NumBlk|:|$S(z)\; =\; \backslash sum\_\{i,j=1\}^n\; z\_i\; z\_j\; h\_\{ij\}(z).$|{{EquationRef|1}}}}

Recall that an arbitrary matrix {{mvar|A}} can be represented as a sum of symmetric {{math|A^(s)}} and anti-symmetric {{math|A^(a)}} matrices,

:$A\_\{ij\}\; =\; A\_\{ij\}^\{(s)\}\; +\; A\_\{ij\}^\{(a)\},\; \backslash qquad\; A\_\{ij\}^\{(s)\}\; =\; \backslash tfrac\{1\}\{2\}\backslash left(\; A\_\{ij\}\; +\; A\_\{ji\}\; \backslash right),\; \backslash qquad\; A\_\{ij\}^\{(a)\}\; =\; \backslash tfrac\{1\}\{2\}\backslash left(\; A\_\{ij\}\; -\; A\_\{ji\}\; \backslash right).$

The contraction of any symmetric matrix B with an arbitrary matrix {{mvar|A}} is

{{NumBlk|:|$\backslash sum\_\{i,j\}\; B\_\{ij\}\; A\_\{ij\}\; =\; \backslash sum\_\{i,j\}\; B\_\{ij\}\; A\_\{ij\}^\{(s)\},$

|{{EquationRef|2}}}}

i.e., the anti-symmetric component of {{mvar|A}} does not contribute because

:$\backslash sum\_\{i,j\}\; B\_\{ij\}\; C\_\{ij\}\; =\; \backslash sum\_\{i,j\}\; B\_\{ji\}\; C\_\{ji\}\; =\; -\; \backslash sum\_\{i,j\}\; B\_\{ij\}\; C\_\{ij\}\; =\; 0.$

Thus, {{math|h_ij(z)}} in equation (1) can be assumed to be symmetric with respect to the interchange of the indices {{mvar|i}} and {{mvar|j}}. Note that

:$\backslash left.\; \backslash frac\{\backslash partial^2\; S(z)\}\{\backslash partial\; z\_i\; \backslash partial\; z\_j\}\; \backslash right|\_\{z=0\}\; =\; 2h\_\{ij\}(0);$

hence, {{math|det(h_ij(0)) ≠ 0}} because the origin is a non-degenerate saddle point.

Let us show by induction that there are local coordinates {{math|u {{=}} (u₁, ... u_n), z {{=}} ψ(u), 0 {{=}} ψ(0)}}, such that

{{NumBlk|:|$S(\backslash boldsymbol\{\backslash psi\}(u))\; =\; \backslash sum\_\{i=1\}^n\; u\_i^2.$|{{EquationRef|3}}}}

First, assume that there exist local coordinates {{math|y {{=}} (y₁, ... y_n), z {{=}} φ(y), 0 {{=}} φ(0)}}, such that

{{NumBlk|:|$S(\backslash boldsymbol\{\backslash phi\}(y))\; =\; y\_1^2\; +\; \backslash cdots\; +\; y\_\{r-1\}^2\; +\; \backslash sum\_\{i,j\; =\; r\}^n\; y\_i\; y\_j\; H\_\{ij\}\; (y),$|{{EquationRef|4}}}}

where {{math|H_ij}} is symmetric due to equation (2). By a linear change of the variables {{math|(y_r, ... y_n)}}, we can assure that {{math|H_rr(0) ≠ 0}}. From the chain rule, we have

:$\backslash frac\{\backslash partial^2\; S\; (\backslash boldsymbol\{\backslash phi\}(y))\}\{\backslash partial\; y\_i\; \backslash partial\; y\_j\}\; =\; \backslash sum\_\{l,k=1\}^n\; \backslash left.\; \backslash frac\{\backslash partial^2\; S\; (z)\}\{\backslash partial\; z\_k\; \backslash partial\; z\_l\}\; \backslash right|\_\{z=\backslash boldsymbol\{\backslash phi\}(y)\}\; \backslash frac\{\backslash partial\; \backslash phi\_k\}\{\backslash partial\; y\_i\}\; \backslash frac\{\backslash partial\; \backslash phi\_l\}\{\backslash partial\; y\_j\}\; +\; \backslash sum\_\{k=1\}^n\; \backslash left.\; \backslash frac\{\backslash partial\; S\; (z)\}\{\backslash partial\; z\_k\; \}\; \backslash right|\_\{z=\backslash boldsymbol\{\backslash phi\}(y)\}\; \backslash frac\{\backslash partial^2\; \backslash phi\_k\}\{\backslash partial\; y\_i\; \backslash partial\; y\_j\}$

Therefore:

:$S$_{yy} (\boldsymbol{\phi}(0)) = \boldsymbol{\phi}'_y(0)^T S_{zz}(0) \boldsymbol{\phi}'_y(0), \qquad \det \boldsymbol{\phi}'_y(0) \neq 0;

whence,

:$0\; \backslash neq\; \backslash det\; S\text{'}\text{'}\_\{yy\}\; (\backslash boldsymbol\{\backslash phi\}(0))\; =\; 2^\{r-1\}\; \backslash det\; \backslash left(\; 2H\_\{ij\}(0)\; \backslash right).$

The matrix {{math|(H_ij(0))}} can be recast in the Jordan normal form: {{math|(H_ij(0)) {{=}} LJL⁻¹}}, were {{mvar|L}} gives the desired non-singular linear transformation and the diagonal of {{mvar|J}} contains non-zero eigenvalues of {{math|(H_ij(0))}}. If {{math|H_ij(0) ≠ 0}} then, due to continuity of {{math|H_ij(y)}}, it must be also non-vanishing in some neighborhood of the origin. Having introduced $\backslash tilde\{H\}\_\{ij\}(y)\; =\; H\_\{ij\}(y)/H\_\{rr\}(y)$, we write

:$\backslash begin\{align\}$

S(\boldsymbol{\varphi}(y)) =& y_1^2 + \cdots + y_{r-1}^2 + H_{rr}(y) \sum_{i,j = r}^n y_i y_j \tilde{H}_{ij} (y) \\

=& y_1^2 + \cdots + y_{r-1}^2 + H_{rr}(y)\left[ y_r^2 + 2y_r \sum_{j=r+1}^n y_j \tilde{H}_{rj} (y) + \sum_{i,j = r+1}^n y_i y_j \tilde{H}_{ij} (y) \right] \\

=& y_1^2 + \cdots + y_{r-1}^2 + H_{rr}(y)\left[ \left( y_r + \sum_{j=r+1}^n y_j \tilde{H}_{rj} (y)\right)^2 - \left( \sum_{j=r+1}^n y_j \tilde{H}_{rj} (y)\right)^2 \right] + H_{rr}(y) \sum_{i,j = r+1}^n y_i y_j \tilde{H}_{ij}(y)

\end{align}

Motivated by the last expression, we introduce new coordinates {{math|z {{=}} η(x), 0 {{=}} η(0),}}

:$x\_r\; =\; \backslash sqrt\{\; H\_\{rr\}(y)\; \}\; \backslash left(\; y\_r\; +\; \backslash sum\_\{j=r+1\}^n\; y\_j\; \backslash tilde\{H\}\_\{rj\}\; (y)\backslash right),\; \backslash qquad\; x\_j\; =\; y\_j,\; \backslash quad\; \backslash forall\; j\; \backslash neq\; r.$

The change of the variables {{math|y ↔ x}} is locally invertible since the corresponding Jacobian is non-zero,

:$\backslash left.\; \backslash frac\{\backslash partial\; x\_r\}\{\backslash partial\; y\_k\}\; \backslash right|\_\{y=0\}\; =\; \backslash sqrt\{H\_\{rr\}(0)\}\; \backslash left[\; \backslash delta\_\{r,\backslash ,\; k\}\; +\; \backslash sum\_\{j=r+1\}^n\; \backslash delta\_\{j,\; \backslash ,\; k\}\; \backslash tilde\{H\}\_\{jr\}(0)\; \backslash right].$

Therefore,

{{NumBlk|:|$S(\backslash boldsymbol\{\backslash eta\}(x))\; =\; \{x\}\_1^2\; +\; \backslash cdots\; +\; \{x\}\_r^2\; +\; \backslash sum\_\{i,j\; =\; r+1\}^n\; \{x\}\_i\; \{x\}\_j\; W\_\{ij\}\; (x).$|{{EquationRef|5}}}}

Comparing equations (4) and (5), we conclude that equation (3) is verified. Denoting the eigenvalues of $S$_{zz}(0) by {{math|μ_j''}}, equation (3) can be rewritten as

{{NumBlk|:|$S(\backslash boldsymbol\{\backslash varphi\}(w))\; =\; \backslash frac\; 12\; \backslash sum\_\{j=1\}^n\; \backslash mu\_j\; w\_j^2.$|{{EquationRef|6}}}}

Therefore,

{{NumBlk|:|$S$_{ww} (\boldsymbol{\varphi}(0)) = \boldsymbol{\varphi}'_w(0)^T S_{zz}(0) \boldsymbol{\varphi}'_w(0),|{{EquationRef|7}}}}

From equation (6), it follows that $\backslash det\; S$_{ww} (\boldsymbol{\varphi}(0)) = \mu_1 \cdots \mu_n. The Jordan normal form of $S$_{zz}(0) reads $S$_{zz}(0) = P J_z P^{-1}, where {{math|J_z}} is an upper diagonal matrix containing the eigenvalues and {{math|det P ≠ 0}}; hence, $\backslash det\; S$_{zz} (0) = \mu_1 \cdots \mu_n. We obtain from equation (7)

:$\backslash det\; S$_{ww} (\boldsymbol{\varphi}(0)) = \left[\det \boldsymbol{\varphi}'_w(0) \right]^2 \det S_{zz}(0) \Longrightarrow \det \boldsymbol{\varphi}'_w(0) = \pm 1.

If $\backslash det\; \backslash boldsymbol\{\backslash varphi\}\text{'}\_w(0)\; =\; -1$, then interchanging two variables assures that $\backslash det\; \backslash boldsymbol\{\backslash varphi\}\text{'}\_w(0)\; =\; +1$.

}}

Assume

1. {{math| f (z)}} and {{math|S(z)}} are holomorphic functions in an open, bounded, and simply connected set {{math|Ω_x ⊂ Cⁿ}} such that the {{math|I_x {{=}} Ω_x ∩ Rⁿ}} is connected;
2. $\backslash Re(S(z))$ has a single maximum: $\backslash max\_\{z\; \backslash in\; I\_x\}\; \backslash Re(S(z))\; =\; \backslash Re(S(x^0))$ for exactly one point {{math|x⁰ ∈ I_x}};
3. {{math|x⁰}} is a non-degenerate saddle point (i.e., {{math|∇S(x⁰) {{=}} 0}} and $\backslash det\; S\text{'}\text{'}\_\{xx\}(x^0)\; \backslash neq\; 0$).

Then, the following asymptotic holds

{{NumBlk|:|$I(\backslash lambda)\; \backslash equiv\; \backslash int\_\{I\_x\}\; f(x)\; e^\{\backslash lambda\; S(x)\}\; dx\; =\; \backslash left(\; \backslash frac\{2\backslash pi\}\{\backslash lambda\}\backslash right)^\{\backslash frac\{n\}\{2\}\}\; e^\{\backslash lambda\; S(x^0)\}\; \backslash left(f(x^0)+\; O\backslash left(\backslash lambda^\{-1\}\backslash right)\; \backslash right)\; \backslash prod\_\{j=1\}^n\; (-\backslash mu\_j)^\{-\backslash frac\{1\}\{2\}\},\; \backslash qquad\; \backslash lambda\; \backslash to\; \backslash infty,$|{{EquationRef|8}}}}

where {{math|μ_j}} are eigenvalues of the Hessian $S\text{'}\text{'}\_\{xx\}(x^0)$ and $(-\backslash mu\_j)^\{-\backslash frac\{1\}\{2\}\}$ are defined with arguments

{{NumBlk|:|

|{{EquationRef|9}}}}

This statement is a special case of more general results presented in Fedoryuk (1987).{{harvtxt|Fedoryuk|1987}}, pages 417-420.

{{math proof|title=Derivation of equation (8)|proof=

File:Illustration To Derivation Of Asymptotic For Saddle Point Integration.pdf

First, we deform the contour {{math|I_x}} into a new contour $I\text{'}\_x\; \backslash subset\; \backslash Omega\_x$ passing through the saddle point {{math|x⁰}} and sharing the boundary with {{math|I_x}}. This deformation does not change the value of the integral {{math|I(λ)}}. We employ the Complex Morse Lemma to change the variables of integration. According to the lemma, the function {{math|φ(w)}} maps a neighborhood {{math|x⁰ ∈ U ⊂ Ω_x}} onto a neighborhood {{math|Ω_w}} containing the origin. The integral {{math|I(λ)}} can be split into two: {{math|I(λ) {{=}} I₀(λ) + I₁(λ)}}, where {{math|I₀(λ)}} is the integral over $U\backslash cap\; I\text{'}\_x$, while {{math|I₁(λ)}} is over $I\text{'}\_x\; \backslash setminus\; (U\backslash cap\; I\text{'}\_x)$ (i.e., the remaining part of the contour {{math|I′_x}}). Since the latter region does not contain the saddle point {{math|x⁰}}, the value of {{math|I₁(λ)}} is exponentially smaller than {{math|I₀(λ)}} as {{math|λ → ∞}};This conclusion follows from a comparison between the final asymptotic for {{math|I₀(λ)}}, given by equation (8), and a simple estimate for the discarded integral {{math|I₁(λ)}}. thus, {{math|I₁(λ)}} is ignored. Introducing the contour {{math|I_w}} such that $U\backslash cap\; I\text{'}\_x\; =\; \backslash boldsymbol\{\backslash varphi\}(I\_w)$, we have

{{NumBlk|:|$I\_0(\backslash lambda)\; =\; e^\{\backslash lambda\; S(x^0)\}\; \backslash int\_\{I\_w\}\; f[\backslash boldsymbol\{\backslash varphi\}(w)]\; \backslash exp\backslash left(\; \backslash lambda\; \backslash sum\_\{j=1\}^n\; \backslash tfrac\{\backslash mu\_j\}\{2\}\; w\_j^2\; \backslash right)\; \backslash left\; |\backslash det\backslash boldsymbol\{\backslash varphi\}\_w\text{'}(w)\; \backslash right\; |\; dw.$|{{EquationRef|10}}}}

Recalling that {{math|x⁰ {{=}} φ(0)}} as well as $\backslash det\; \backslash boldsymbol\{\backslash varphi\}\_w\text{'}(0)\; =\; 1$, we expand the pre-exponential function $f[\backslash boldsymbol\{\backslash varphi\}(w)]$ into a Taylor series and keep just the leading zero-order term

{{NumBlk|:|$I\_0(\backslash lambda)\; \backslash approx\; f(x^0)\; e^\{\backslash lambda\; S(x^0)\}\; \backslash int\_\{\backslash mathbf\{R\}^n\}\; \backslash exp\; \backslash left(\; \backslash lambda\; \backslash sum\_\{j=1\}^n\; \backslash tfrac\{\backslash mu\_j\}\{2\}\; w\_j^2\; \backslash right)\; dw\; =\; f(x^0)e^\{\backslash lambda\; S(x^0)\}\; \backslash prod\_\{j=1\}^n\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; e^\{\backslash frac\{1\}\{2\}\backslash lambda\; \backslash mu\_j\; y^2\}\; dy.$|{{EquationRef|11}}}}

Here, we have substituted the integration region {{math|I_w}} by {{math|Rⁿ}} because both contain the origin, which is a saddle point, hence they are equal up to an exponentially small term.This is justified by comparing the integral asymptotic over {{math|Rⁿ}} [see equation (8)] with a simple estimate for the altered part. The integrals in the r.h.s. of equation (11) can be expressed as

{{NumBlk|:|$\backslash mathcal\{I\}\_j\; =\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; e^\{\backslash frac\{1\}\{2\}\; \backslash lambda\; \backslash mu\_j\; y^2\}\; dy\; =\; 2\backslash int\_0^\{\backslash infty\}\; e^\{-\backslash frac\{1\}\{2\}\; \backslash lambda\; \backslash left(\backslash sqrt\{-\backslash mu\_j\}\; y\backslash right)^2\}\; dy\; =\; 2\backslash int\_0^\{\backslash infty\}\; e^\{-\backslash frac\{1\}\{2\}\; \backslash lambda\; \backslash left\; |\backslash sqrt\{-\backslash mu\_j\}\; \backslash right|^2\; y^2\backslash exp\backslash left(2i\backslash arg\backslash sqrt\{-\backslash mu\_j\}\backslash right)\}\; dy.$|{{EquationRef|12}}}}

From this representation, we conclude that condition (9) must be satisfied in order for the r.h.s. and l.h.s. of equation (12) to coincide. According to assumption 2, $\backslash Re\; \backslash left(\; S\_\{xx\}\text{'}\text{'}(x^0)\; \backslash right)$ is a negatively defined quadratic form (viz., ) implying the existence of the integral $\backslash mathcal\{I\}\_j$, which is readily calculated

:$\backslash mathcal\{I\}\_j\; =\; \backslash frac\{2\}\{\backslash sqrt\{-\backslash mu\_j\}\backslash sqrt\{\backslash lambda\}\}\; \backslash int\_0^\{\backslash infty\}\; e^\{-\backslash frac\{\backslash xi^2\}\{2\}\}\; d\backslash xi\; =\; \backslash sqrt\{\backslash frac\{2\backslash pi\}\{\backslash lambda\}\}\; (-\backslash mu\_j)^\{-\backslash frac\{1\}\{2\}\}.$

}}

Equation (8) can also be written as

{{NumBlk|:|$I(\backslash lambda)\; =\; \backslash left(\; \backslash frac\{2\backslash pi\}\{\backslash lambda\}\backslash right)^\{\backslash frac\{n\}\{2\}\}\; e^\{\backslash lambda\; S(x^0)\}\; \backslash left\; (\; \backslash det\; (-S\_\{xx\}\text{'}\text{'}(x^0))\; \backslash right\; )^\{-\backslash frac\{1\}\{2\}\}\; \backslash left\; (f(x^0)\; +\; O\backslash left(\backslash lambda^\{-1\}\backslash right)\; \backslash right),$|{{EquationRef|13}}}}

where the branch of

:$\backslash sqrt\{\backslash det\; \backslash left\; (-S\_\{xx\}\text{'}\text{'}(x^0)\; \backslash right)\}$

is selected as follows

:$\backslash begin\{align\}$

\left (\det \left (-S_{xx}(x^0) \right ) \right)^{-\frac{1}{2}} &= \exp\left( -i \text{ Ind} \left (- S_{xx}(x^0) \right ) \right) \prod_{j=1}^n \left| \mu_j \right|^{-\frac{1}{2}}, \\

\text{Ind} \left (-S_{xx}''(x^0) \right) &= \tfrac{1}{2} \sum_{j=1}^n \arg (-\mu_j), && |\arg(-\mu_j)| < \tfrac{\pi}{2}.

\end{align}

Consider important special cases:

• If {{math|S(x)}} is real valued for real {{mvar|x}} and {{math|x⁰}} in {{math|Rⁿ}} (aka, the multidimensional Laplace method), thenSee equation (4.4.9) on page 125 in {{harvtxt|Fedoryuk|1987}} $$\backslash text\{Ind\}\; \backslash left(-S\_\{xx\}\text{'}\text{'}(x^0)\; \backslash right\; )\; =\; 0.$$
• If {{math|S(x)}} is purely imaginary for real {{mvar|x}} (i.e., $\backslash Re(S(x))\; =\; 0$ for all {{mvar|x}} in {{math|Rⁿ}}) and {{math|x⁰}} in {{math|Rⁿ}} (aka, the multidimensional stationary phase method),Rigorously speaking, this case cannot be inferred from equation (8) because the second assumption, utilized in the derivation, is violated. To include the discussed case of a purely imaginary phase function, condition (9) should be replaced by $\backslash left\; |\; \backslash arg\backslash sqrt\{-\backslash mu\_j\}\; \backslash right\; |\; \backslash leqslant\; \backslash tfrac\{\backslash pi\}\{4\}.$ thenSee equation (2.2.6') on page 186 in {{harvtxt|Fedoryuk|1987}} $$\backslash text\{Ind\}\; \backslash left\; (-S\_\{xx\}$$(x^0) \right ) = \frac{\pi}{4} \text{sign }S_{xx}(x_0), where $\backslash text\{sign\; \}S\_\{xx\}$(x_0) denotes the signature of matrix $S\_\{xx\}$(x_0), which equals to the number of negative eigenvalues minus the number of positive ones. It is noteworthy that in applications of the stationary phase method to the multidimensional WKB approximation in quantum mechanics (as well as in optics), {{math|Ind}} is related to the Maslov index see, e.g., {{harvtxt|Chaichian|Demichev|2001}} and {{harvtxt|Schulman|2005}}.

If the function {{math|S(x)}} has multiple isolated non-degenerate saddle points, i.e.,

:$\backslash nabla\; S\; \backslash left\; (x^\{(k)\}\; \backslash right\; )\; =\; 0,\; \backslash quad\; \backslash det\; S\text{'}\text{'}\_\{xx\}\; \backslash left\; (x^\{(k)\}\; \backslash right\; )\; \backslash neq\; 0,\; \backslash quad\; x^\{(k)\}\; \backslash in\; \backslash Omega\_x^\{(k)\},$

where

:$\backslash left\; \backslash \{\; \backslash Omega\_x^\{(k)\}\; \backslash right\; \backslash \}\_\{k=1\}^K$

is an open cover of {{math|Ω_x}}, then the calculation of the integral asymptotic is reduced to the case of a single saddle point by employing the partition of unity. The partition of unity allows us to construct a set of continuous functions {{math|ρ_k(x) : Ω_x → [0, 1], 1 ≤ k ≤ K,}} such that

:$\backslash begin\{align\}$

\sum_{k=1}^K \rho_k(x) &= 1, && \forall x \in \Omega_x, \\

\rho_k(x) &= 0 && \forall x \in \Omega_x\setminus \Omega_x^{(k)}.

\end{align}

Whence,

:$\backslash int\_\{I\_x\; \backslash subset\; \backslash Omega\_x\}\; f(x)\; e^\{\backslash lambda\; S(x)\}\; dx\; \backslash equiv\; \backslash sum\_\{k=1\}^K\; \backslash int\_\{I\_x\; \backslash subset\; \backslash Omega\_x\}\; \backslash rho\_k(x)\; f(x)\; e^\{\backslash lambda\; S(x)\}\; dx.$

Therefore as {{math|λ → ∞}} we have:

:$\backslash sum\_\{k=1\}^K\; \backslash int\_\{\backslash text\{a\; neighborhood\; of\; \}x^\{(k)\}\}\; f(x)\; e^\{\backslash lambda\; S(x)\}\; dx\; =\; \backslash left(\backslash frac\{2\backslash pi\}\{\backslash lambda\}\backslash right)^\{\backslash frac\{n\}\{2\}\}\; \backslash sum\_\{k=1\}^K\; e^\{\backslash lambda\; S\; \backslash left\; (x^\{(k)\}\; \backslash right\; )\}\; \backslash left\; (\; \backslash det\; \backslash left(-S\_\{xx\}\text{'}\text{'}\; \backslash left\; (x^\{(k)\}\; \backslash right\; )\backslash right)\; \backslash right)^\{-\backslash frac\{1\}\{2\}\}\; f\; \backslash left\; (x^\{(k)\}\; \backslash right\; ),$

where equation (13) was utilized at the last stage, and the pre-exponential function {{math| f (x)}} at least must be continuous.

When {{math|∇S(z⁰) {{=}} 0}} and $\backslash det\; S$_{zz}(z^0) = 0, the point {{math|z⁰ ∈ Cn}} is called a degenerate saddle point of a function {{math|S(z'')}}.

Calculating the asymptotic of

:$\backslash int\; f(x)\; e^\{\backslash lambda\; S(x)\}\; dx,$

when {{math|λ → ∞,  f (x)}} is continuous, and {{math|S(z)}} has a degenerate saddle point, is a very rich problem, whose solution heavily relies on the catastrophe theory. Here, the catastrophe theory replaces the Morse lemma, valid only in the non-degenerate case, to transform the function {{math|S(z)}} into one of the multitude of canonical representations. For further details see, e.g., {{harvtxt|Poston|Stewart|1978}} and {{harvtxt|Fedoryuk|1987}}.

Integrals with degenerate saddle points naturally appear in many applications including optical caustics and the multidimensional WKB approximation in quantum mechanics.

The other cases such as, e.g., {{math| f (x)}} and/or {{math|S(x)}} are discontinuous or when an extremum of {{math|S(x)}} lies at the integration region's boundary, require special care (see, e.g., {{harvtxt|Fedoryuk|1987}} and {{harvtxt|Wong|1989}}).

An extension of the steepest descent method is the so-called nonlinear stationary phase/steepest descent method. Here, instead of integrals, one needs to evaluate asymptotically solutions of Riemann–Hilbert factorization problems.

Given a contour C in the complex sphere, a function f defined on that contour and a special point, say infinity, one seeks a function M holomorphic away from the contour C, with prescribed jump across C, and with a given normalization at infinity. If f and hence M are matrices rather than scalars this is a problem that in general does not admit an explicit solution.

An asymptotic evaluation is then possible along the lines of the linear stationary phase/steepest descent method. The idea is to reduce asymptotically the solution of the given Riemann–Hilbert problem to that of a simpler, explicitly solvable, Riemann–Hilbert problem. Cauchy's theorem is used to justify deformations of the jump contour.

The nonlinear stationary phase was introduced by Deift and Zhou in 1993, based on earlier work of the Russian mathematician Alexander Its. A (properly speaking) nonlinear steepest descent method was introduced by Kamvissis, K. McLaughlin and P. Miller in 2003, based on previous work of Lax, Levermore, Deift, Venakides and Zhou. As in the linear case, steepest descent contours solve a min-max problem. In the nonlinear case they turn out to be "S-curves" (defined in a different context back in the 80s by Stahl, Gonchar and Rakhmanov).

The nonlinear stationary phase/steepest descent method has applications to the theory of soliton equations and integrable models, random matrices and combinatorics.

Another extension is the Method of Chester–Friedman–Ursell for coalescing saddle points and uniform asymptotic extensions.

• Pearcey integral
• Stationary phase approximation
• Laplace's method

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Category:Asymptotic analysis

Category:Perturbation theory