QuantumPhysics.dvi

Here we have absorbed various constants (such as ̄h,m) intoωandλ. The integralI(ω,λ)

retains just the basic structure of the path integral. Expanding the integrand in powers of

λfor fixedωgives,

I(ω,λ) =

∑∞

n=0

(−λ)n

n!

∫+∞

−∞

dq q^4 nexp{−ω^2 q^2 } (10.9)

The calculation of the individual integrals for givennare carried out by settingx=q^2 ,

which reduces the integral to a Γ-function. These functions are defined by the integral

representation

Γ(z) =

∫∞

0

dxxz−^1 e−x (10.10)

for all complexzsuch that Re(z)>0. There, the function obeys

Γ(z+ 1) =zΓ(z) (10.11)

Using this relation throughout the plane allows one to analytically continue Γ(z) throughout

the complex plane. The resulting function is holomorphic throughoutthe complex plane,

except for simple poles at all negative integers, and at zero. On positive integersn, the Γ

function reduces to the factorial, Γ(n+1) =n!. Just as for the factorial, the largezbehavior

of the Γ-function is given by Sterling’s formula,

Γ(z) =ezlnz−z

√

2 π

z

{

1 +

1

12 z

+

1

288 z^2

+···

}

|arg(z)|< π (10.12)

For curiosity’s sake, we also expandI(ω,λ) for fixedλand smallω, and using the same

techniques as above, we find,

I(ω,λ) =

1

ω

∑∞

n=0

Γ

(

2 n+^12

)

n!

(

−λ

ω^4

)n

smallλ

I(ω,λ) =

1

4 λ^1 /^4

∑∞

n=0

Γ

( 2 n+1

4

)

n!

(

−ω^2

√

λ

)n

smallω (10.13)

Using Sterling’s formula one may determine the radii of convergenceof each of the above se-

ries expansions. The expansion for fixedλ >0 and smallωhas infinite radius of convergence.

The series for fixedωand smallλ, however, has zero radius of convergence!

The interpretation of this result is very illuminating. Suppose we changed the sign of

λand made λ <0. The integral is then clearly divergent (at q → ∞). If the integral

admitted a Taylor series expansion aroundλ= 0 withfinite radius of convergence, then