A Short Introduction to Method of Steepest Descent

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By : Nadeem Al-Shawaf

2011

Nadeem Al-Shawaf TMS

27/12/2011

Mathematical Methods of Physics

Here you can find a short & non-rigorous motivation and introduction to the

method of steepest descent that is very important in evaluating integrals of

exponent-like function, also you will find two detailed examples that will show how

to use this method to find asymptotic expansion of Airy & Bessel functions.

For questions or comments, email me to tms2006 on gmail.com

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Contents

1. Introduction to Laplace Method .................................................................................................................. 2

2. Watson Lemma ............................................................................................................................................ 2

3. The Asymptotic expansion (Generalization in Real Plane) .......................................................................... 3

4. Extending to Complex Plain; Method of steepest descent, and Stationary Phase method ......................... 4

5. Jordan Lemma .............................................................................................................................................. 6

6. Stokes Phenomena ....................................................................................................................................... 7

7. Airy Functions ............................................................................................................................................... 7

8. Bessel Functions ......................................................................................................................................... 10

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1. Introduction to Laplace Method In 1774 and while Laplace was trying to estimate what we call now inverse Laplace transformation, or integrals of

the form:

( ) ∫ ( ) ( )

Usually when we want to get asymptotic description for such a function, we integrate it by parts then we suppose

that , or:

( ) ∫ ( )

( ) ( ) ( )

∫ ( )

( ) ( )

But as we note if our ( ) has a stationary point ( ) in the domain of integration, we can’t use this method

anymore.

So Laplace made the following observation: when then it’s easy to see that the exponential part will be the

bigger contributor in the integral value, and more precisely the point where ( ) reaches its maximum value in

( ), so if we suppose that ( ) has a stationary point in this interval, we can expand it around this point :

( ) ( )

( )

( )

(We supposed that is a global unique maximum in ( ) so ( ) ) then we can say that our integral is

approximately equal to the same integral in the domain , - were is a sufficiently small number ,

so we can write:

( ) ∫ ( ) ( )

( ) ( ) ∫

( )

( )

The last integral can be reduced to the famous gauss integral as follows:

∫ ( )

√ ( )∫

√

√

√ ( )

∫

√

√

√

( )

It is important to sympathize that these results taking in account that is not equal to one of the integration

bounds, in other case we will need to take a slightly different approach, anyway we do not need that case here.

So finally, we have:

∫ ( ) ( )

( ) ( ) √

( ) {

( )

( )

Also it should be noted, that we suppose that ( ) have only one maximum in , - , the same thing will be

supposed along the rest of the text , anyway it’s obvious that if there is couple such a points, we can simply sum the

asymptotic expansion “around” each of those points.

2. Watson Lemma Watson was trying to find estimation for usual Laplace transform, or in general, he analyzed the integral:

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( ) ∫ ( )

Now the biggest contribution will be done by ( ) instead of the exponential part, so In 1918 Watson suggested to

expand this function ( ) around 0, what gives us:

( ) ∫ ( )

∑ ( )( )

( )

( ) ∫

We immediately note that the last integral is very similar to Gamma function definition (except the upper bound of

the integral) , so we can write:

( ) ∫

∫

( )

( | |) {

( )

Where in we take the main branch (the same will be assumed for all powers in this text).

With a more careful analysis, and after extending the results to complex plane (see below title for that) it can be

shown that this will only add some conditions (beside analyticity of ( ) along integration path ) but doesn’t

change the expansion terms themselves), and we will get that:

∫ ( )

( )

∑ ( )( )

( )

{

( ) ( )

| |

( )

Some notes:

I. It should be noted that the expansion stays valid even if ( ) has singularities as long as the last condition

holds.

II. In addition, I will mention that Watson’s lemma error is of magnitude (| |

) where is the number

of terms that will be taken in the above sum. Anyway it is still asymptotic expansion what means that this

error holds up to a specific only.

III. Some authors of mathematical physics, considers and build the above result by expanding ( ( )) by

powers of √ directly, even so it’s gives the same results, as I understood, from mathematical rigor point of

view, it’s not quite right because in case is complex or negative, it will need a different approach.

3. The Asymptotic expansion (Generalization in Real Plane)

Laplace method actually is quite rough estimation, and here comes the real Importance of Watson lemma, actually

the main integral ( ) can be converted to an integral of the form ( ) that used in Watson Lemma, by a tricky

substitution, if we suppose that:

( ) ( ) ( )( ) ( )( )

Then if we make a substitution, ( ) where the last function is chosen to satisfy (it can be proven that this

possible using Lagrange inversion theorem):

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, ( )- ( ) {

( )

( ) √( )

| ( )( )|

(last expression shows that we moved the saddle point away) , Then we will have:

( ) ( ) ∫ * , ( )- ( )+

Now by Appling Watson lemma it’s easy to get that (also already it will include the complex functions case, see the

next paragraph):

∫ ( ) ( )

( )

( )

√ ∑

( )

( )

(

)

( )

{

( )

, ( ) ( )-

( )

( )

( ), ( ) ( )-|

( ) ( ) , ( )-

( )( ) ( )( )

| |

| |

| ( ) ( )|

Where ( ) means a neighborhood to of radius.

4. Extending to Complex Plain (Method of steepest descent, and Stationary Phase method) In 1909, Debye while he was trying to find asymptotic approximation for Bessel functions; he proposed an

extension for Laplace method on complex plane (he did that depending on Old Riemann notes in 1863).

For the beginning we consider and ideal situation when both of ( ) ( ) are analytical in some open domain that

includes integration path and has no singularities inside, while there is only one saddle point for which ( )

and it is not positioned on the end or the beginning of integration path , and we want to evaluate the integral:

( ) ∫ ( ) ( )

( )

In a same way as Laplace discussed, the exponential part is the most important contributor, and according to

Cauchy integral theorem, and because the integrant is analytical, we can:

I. Deform the integrating path to give it a shape that is most comfortable for us.

II. When we do that we should keep in mind that both of the ending points of this path (if any) should be

fixed.

III. Deformation should be done in a way that ensures that we will not drug the integrant over any

singularities.

IV. Finally, we need it to go through the saddle point as will be explained below.

By doing that the approximated value of our integral doesn’t change (we will denote the set of all such paths ).

It should be noted that it may sounds that deforming a path that is already on real axis will make the problem more

complex, but actually we may find in complex plane a “better” saddle points that will make integrant grows faster

and will make our estimation “better”, this makes the process of choosing the saddle point that we will go through

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seems to be very subjective, but as we will see below, there will be a specific properties to the path and saddle

point that we will choose.

Let’s suppose that If ( ) ( ) ( ) then we can rewrite our integral in the form:

( ) ∫ ( ) ( ) ( )

( )

So to apply Laplace method, we need to make the amplitude part ( ) to increase at maximum possible speed

along our path (that going through the saddle point), while the phase part ( ) (if where complex, obviously

we should discuss behavior of * , ( )-+ instead, and the same is for the previous term) to change at

minimum possible speed (this is to decrease the oscillation effect of the imaginary part when we approaching the

saddle point).

The Main problem in applying the Real Integral case to complex plane that

behaves in different ways when it approaching it’s saddle point from different

directions, because we are now moving on a manifold not in a flat plane as in Real

Integral case (see the figure), this makes the saddle point looks like a hill when we

moving on one path, and like a valley on another.

Second problem, is to make the Phase part oscillates to minimum possible value to

decrease its contribution on estimation.

Fortunately, analytical complex functions has some futures that can in second issue, so Debye as a physicists knew

that complex vector field is conservative, what means that level curves (equipotential in physical terms) of

imaginary and real parts of any Meromorphic complex function will be orthogonal; this easily can be seen as

follows:

If ( ) ( ) ( ) then it’s clear from Cauchy-Riemann equations that but from gradient

definition we know that ( ) ( ) , that means if we will move along

imaginary equipotential, the real part of the function will grow at maximum possible speed and vice versa, and this

exactly what we were looking for to apply Laplace method, what solves the second problem.

I found the following two nice graphs that illustrate this concept very good:

( ) ( ) Two graphs overlapped

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What we get from this, is if we ask to satisfy , ( )- , this will ensure us that at we will have a set of

stationary points (this condition is not enough to determine them) that will make our amplitude part of the

integrant maximized, in the same time , if we ask to have , ( )- , ( )- what will ensure

that the real part will increase at it’s maximum speed as discussed above, and the phase part is not oscillating , but

those both conditions combined is ( ) what means that will be a saddle point, this approach is what

called the “Method of Steepest Descent”, in some other cases we may be interested to make , ( )-

, ( )- , then it will be called “Method of Stationary Phase” the one I will not talk about here.

Off course this doesn’t solve the first problem, usually we overridden simply by making sure that , ( )-

, ( )- along all our chosen path (up to infinity if any), this will ensure that our stationary point actually is

global and that our path is mountain-like.

More precisely, Mathematically, the best possible path should go thought a point such as:

| ( ) ( )|

But unfortunately, in general , there is no precise way to find such a path, and in some cases (even when ( ) is

simple) there is no such a path at all, but even so, if we success in finding it (we will call it ) we can convert our

complex integral to a real integral on some peace of our path near the saddle point (in analog to when make

chanced integration bounds in Laplace method to be just near the stationary point) , by some substitution along

this path, and then simply we can apply original Laplace method or Watson Lemma:

Same as we did in the previous paragraph, If we put ( ) ( ) and because

( )

( )

|

√

( )

(as usually we taking the main brunch of the roots) We get that:

( ) ∫ ( ) ( )

( )

( ) ∫ , ( )-

For which we can now use Watson lemma.

Anyway, to have all this be right, we at first usually should make sure that our Integral coverage’s, this usually easy

to be done by Jordan Lemma…

5. Jordan Lemma It’s a known and very useful lemma to give a quick estimation for integrals, and it states:

| ∫ ( )

( )

|

, -

| ( )|

Where is an upper half circle of radius R that aligned on Real axis and its center in the center of coordinates.

It’s main application that if | ( )| and ( ) has no purely real singularities, then we can calculate

easily the real integral along real axis which is actually the radius of the above half circle, or:

∫ ( )

∑ ( )

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6. Stokes Phenomena When steepest descent path of integration in complex plane happens to have for example two saddle points on his

way, and if , it may be that the contribution of each saddle radically changes for different (such as

changes to ) this leads to what called Stokes Phenomena, that forces to have different asymptotic expansions on

different regions of complex plane.

The above reason is not the only one for this phenomena, it can happen in a more obvious situations like if ( ) is

not single valued function, or even in a less obvious situations like in the case of “complementary error function":

( ) ( )

√ ∫

{

( ) | |

( )

| |

( )

| |

( )

√

{

(

)}

As we see that some sectors are also overlapped, and the same sector may have different expansions.

This phenomenon cases a serious problems in commutability & drawing of differential equations solutions, and

currently is under extensive research.

A more interesting behavior of this phenomenon will be mentioned in the end of the Airy’s functions.

7. Airy Functions During His research in 1838 in Optical diffraction (anyway they can be found in wave theory, electron tunneling and

other fields, or scale-invariant form of the Schrodinger equation for a quantum particle under a fixed potential) he

faced a need to solve equation of the form:

( ) ( )

It’s solution is a linear combination of two linearly independent functions, Airy’s of the first and second type.

It’s asymptotic expansion can be found in couple ways, by steepest descent or stationary phase methods, and in

both cases if we consider that we know the analytical extension to complex plane of Airy’s function the question

became much easier, also it should be noted that because Airy Functions are a special case of Bessel functions this

adds more variety on the methods that one can use.

As for me, I Think the best way to understand how to make the approximation is to understand how we got the

solution of the equation itself, otherwise all tries will unfruitful.

Let’s consider the following inverse Laplace transform of our solution:

( ) , ( )- ∫ ( )

( )

Where ( ) is analytical function without any singularites, and is a counter that do not depend on and will be

determine latter, and If we put the boundary condition ( )| it can be easily shown by integration by

parts that:

( ) ∫ ( )

( )

( )

Now let’s discuss integration contour, if it was a finite and closed contour, it will give us a trivial zero solution

because the integrant has no singularities (Cauchy's integral theorem), while taking a finite unclosed contour will

violate the boundary condition we put above, thus it should be infinite across the whole complex plane.

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Now to make sure that our integral will converge, we have to make sure that the integrant is decreasing while we

approaching infinity, and that can happen only in three sectors; because:

| | , ( )- ( ) | |

{ ( )

, ( )-

{

0

1

[

]

[

]

But because all of this sectors are equivalent, and our contour should go from infinity

to infinity, we can suppose that our counter should go through any of the two sectors,

what means that we will get 3 different paths as illustrated in the figure, so

we can consider 3 solutions for each of those counters.

But, interestingly, we can see that if we cycle along all our three paths we will get a

closed counter, then the integral along according to Cauchy vanishes as

discussed above, what means that ( ) ( ) ( ) thus, actually we have

only two independent solutions! And they called Airy’s functions of First and second

kind (or regular and irregular), and defined as follows:

( )

∫

( )

( )

∫

( )

It should be noted that (as it will be showed below) we can take to be along the

imaginary axis only, and If we put , then by Jordan Lemma we still have

a coverage integral that can be written as:

( )

∫ (

)

∫ (

)

And the last integral is actually the original form Airy worked with, some authors starting with it when trying to find

it’s estimation, but I found it to be very unnatural way and preferred to build it from ground.

Now we need to find the best path of integration for approximation, first to make things easier we can shrink the

scale by putting √ and we get:

( ) √ ∫

( )

( )

( ) √

∫ ( )

( )

( )

The integration counters will not change its shape because we just shrink our coordinates.

Then the saddle points are:

( )

( )

So we have actually a circle of saddle points, and it’s clear that they will affect our condition for steepest descent

path.

let’s for beginning choose the easiest way and put , then we see that it’s not straight forward do deform

to go through , while it’s a natural thing to do for , so we will consider this saddle for the Airy’s second

kind functions, and for the first kind that we will work on.

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As discussed before, for our steepest descent path we have to satisfy the condition [ ( ) ( )] as

follows:

( ) ( ) ( )

( )

( ) ( ) ( ) [ ( ) ( )] (

)

So this will be satisfied in two sectors (see the magnification of the figure) because:

( ) { [

]

[

]

Those sectors are showed also in the magnification of the figure, now as discussed above; to decrease oscillations

we need to make:

, ( )- , ( )- ( )

What means that our path near the saddle point will be simply parallel to imaginary axis and that will guaranty

maximizing the amplitude part and vanishing the phase part.

Now let’s analysis how may affect the results, to satisfy the needed path as in the figure we take but

we showed that the condition ( ) so from here we can easily find that this will be satisfied if

| | | | , in this case it will not chage the phase part begaviour, so we can see here that Stockes

phenomena may has its influence when this condition violated.

So everything ready to use the above-mentioned extension to complex plane and by using Watson lemma it’s

straight away to get that:

( ) ⁄

√ ∑

( )

( ) (

)

| |

Where all roots are to be taken as the main brunch.

Now let’s see the other case, or when , and of course to simplify things we take ,

by direct substitution into the result we got above we get:

( )

( )

The most important change happened here is that [ ( )] what means in its turn that in this case the both

saddle points should contribute in the asymptotic expansion because they have the same weight, and it’s easy to

see how to deform to go from some complex infinity to then to move along imaginary axis again to and to

infinity again, and verifying all the needed sectors as we did above will tell us that this path is valid, so basically we

apply the same steps and we will get that:

( ) ⁄

√ ∑

( ) (

)

| |

And another important difference in this case, that we should take the roots as: √ |√

| √ |√ |.

As we see, this is another sample of Stockes phenomena, and actually, if we were analyzed asymptotic expansion

for ( ) , we will find that those expansion’s values in specific sectors exchanges with ( ) expansion! In other

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words expansion for ( ) for some values will give the other solution of the differential equation which is ( )

, this mutability actually arises because the steepest descent path changes discontinuously relative to in some

positions.

8. Bessel Functions The situation with Bessel functions is much more complicated compared to Airy’s , because there is different

integral representation for them and different modification of these function, also the expansion of ( ) is differs

for large and small one, for large arguments and small one, for Integer, half integer, Purely Imaginary

…etc. arguments, all of them has different treatment and asymptotic behavior, also new methods are presented till

current days, so we will consider the real arguments here.

Bessel functions are the solution of the following differential equation:

( )

Now we need to find it’s integral representation in the complex plan, below is somehow a clumsy way to do that

but it’s true anyway.

Considering our knowledge that Bessel functions are actually orthogonal functions, we can suppose that we can do

the following furrier expansion:

( ) ∑

( )

We can make sure of that by simply bulging it into the diff. equation.

Now we note that if we multiply the both sides by and integrate them we get:

∫ . /

( )

∑ ( ) ∫

( )

We note that if our integration contour will enclose the origin, then by reside theorem we get that:

∫

( )

( is Kronecker delta) (actually we can expand set to be more than just natural numbers but it’s not our

concern here) thus we found the following integral representation for Bessel functions of the first kind:

( )

∫ ( )

( )

Basically we have some freedom in choosing our path, we can take it simplify as

a unit circle around the origin or as illustrated in the picture, we will take the

first, so we can put (this choice is comfortable because it will cancel

out some terms in the calculations below).

So our saddle points are:

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( ) ( )

(

)

And it’s quite obvious how to deform path to go through the saddle points, also I’s clear that both of the points will

contribute in the expansion.

Now if we use a new integration parameter such as:

( ) ( ) ( )

√

Where we choice square root sign in such a way that will keep the direction of our path counterclockwise, also we

see how our will cancel out from the exponent and no more effect our steepest descent path, and our integration

path near saddle points will as showed in by arrows in the above figure.

Now everything is ready to apply Watson lemma for each saddle point will give us:

( ) {

√ ∑

( )

} {

√ ∑

( )

}

Where:

{

( )

( ) ∏, ( ) -

( )

( ) ∏, ( ) -

And this expansion is valid for any , anyway the asymptotic actually is better in case | | | |

otherwise it should be expanded little bit differently.

Documents

A Short Introduction to Method of Steepest Descent