Method of Chester–Friedman–Ursell

In asymptotic analysis, the method of Chester–Friedman–Ursell is a technique to find asymptotic expansions for contour integrals. It was developed as an extension of the steepest descent method for getting uniform asymptotic expansions in the case of coalescing saddle points. The method was published in 1957 by Clive R. Chester, Bernard Friedman and Fritz Ursell.

Method

Setting

We study integrals of the form

I

(

α

,

N

)

:=

∫

C

e

−

N

f

(

α

,

t

)

g

(

α

,

t

)

d

t

,

{\displaystyle I(\alpha ,N):=\int _{C}e^{-Nf(\alpha ,t)}g(\alpha ,t)dt,}

where

C

{\displaystyle C}

is a contour and

f

,

g

{\displaystyle f,g}

are two analytic functions in the complex variable

t

{\displaystyle t}

and continuous in

α

{\displaystyle \alpha }

.

N

{\displaystyle N}

is a large number.

Suppose we have two saddle points

t

+

,

t

−

{\displaystyle t_{+},t_{-}}

of

f

(

α

,

t

)

{\displaystyle f(\alpha ,t)}

with multiplicity

1

{\displaystyle 1}

that depend on a parameter

α

{\displaystyle \alpha }

. If now an

α

0

{\displaystyle \alpha _{0}}

exists, such that both saddle points coalescent to a new saddle point

t

0

{\displaystyle t_{0}}

with multiplicity

2

{\displaystyle 2}

, then the steepest descent method no longer gives uniform asymptotic expansions.

Procedure

Suppose there are two simple saddle points

t

−

:=

t

−

(

α

)

{\displaystyle t_{-}:=t_{-}(\alpha )}

and

t

+

:=

t

+

(

α

)

{\displaystyle t_{+}:=t_{+}(\alpha )}

of

f

{\displaystyle f}

and suppose that they coalescent in the point

t

0

:=

t

0

(

α

0

)

{\displaystyle t_{0}:=t_{0}(\alpha _{0})}

.

We start with the cubic transformation

t

↦

w

{\displaystyle t\mapsto w}

of

f

(

α

,

t

)

{\displaystyle f(\alpha ,t)}

, this means we introduce a new complex variable

w

{\displaystyle w}

and write

f

(

α

,

t

)

=

1

3

w

3

−

η

(

α

)

w

+

A

(

α

)

,

{\displaystyle f(\alpha ,t)={\tfrac {1}{3}}w^{3}-\eta (\alpha )w+A(\alpha ),}

where the coefficients

η

:=

η

(

α

)

{\displaystyle \eta :=\eta (\alpha )}

and

A

:=

A

(

α

)

{\displaystyle A:=A(\alpha )}

will be determined later.

We have

d

t

d

w

=

w

2

−

η

f

t

(

α

,

t

)

,

{\displaystyle {\frac {dt}{dw}}={\frac {w^{2}-\eta }{f_{t}(\alpha ,t)}},}

so the cubic transformation will be analytic and injective only if

d

t

/

d

w

{\displaystyle dt/dw}

and

d

w

/

d

t

{\displaystyle dw/dt}

are neither

0

{\displaystyle 0}

nor

∞

{\displaystyle \infty }

.