2—Infinite Series 31Differentiate thismtimes with respect toxandntimes with respect toy, then setx=aandy=b.
Only one term survives and that is
∂m+nf
∂xm∂yn
(a,b) =m!n!Amn
I can use subscripts to denote differentiation so that ∂f∂x isfxand ∂
(^3) f
∂x^2 ∂y isfxxy. Then the
two-variable Taylor expansion is
f(x,y) =f(0)+fx(0)x+fy(0)y+
1
2
[
fxx(0)x^2 + 2fxy(0)xy+fyy(0)y^2
]
+
1
3!
[
fxxx(0)x^3 + 3fxxy(0)x^2 y+ 3fxyy(0)xy^2 +fyyy(0)y^3
]
+··· (2.16)
Again put more order into the notation and rewrite the general form usingAmnas
Amn=
1
(m+n)!
(
(m+n)!
m!n!
)
∂m+nf
∂xm∂yn
(a,b) (2.17)
That factor in parentheses is variously called the binomial coefficient or a combinatorial factor. Standard
notations for it are
m!
n!(m−n)!
=mCn=
(
m
n
)
(2.18)
The binomial series, Eq. (2.4), for the case of a positive integer exponent is
(1 +x)m=
∑mn=0(
m
n
)
xn, or more symmetrically
(a+b)m=
∑mn=0(
m
n
)
anbm−n (2.19)
(a+b)^2 =a^2 + 2ab+b^2 , (a+b)^3 =a^3 + 3a^2 b+ 3ab^2 +b^3 ,
(a+b)^4 =a^4 + 4a^3 b+ 6a^2 b^2 + 4ab^3 +b^4 , etc.
Its relation to combinatorial analysis is that if you ask how many different ways can you choosen
objects from a collection ofmof them,mCnis the answer.
2.6 Stirling’s Approximation
The Gamma function for positive integers is a factorial. A clever use of infinite series and Gaussian
integrals provides a useful approximate value for the factorial of largen.
n!∼
√
2 πnnne−n for largen (2.20)
Start from the Gamma function ofn+ 1.
n! = Γ(n+ 1) =
∫∞
0dttne−t=
∫∞
0dte−t+nlnt
The integrand starts at zero, increases, and drops back down to zero ast→ ∞. The graph roughly
resembles a Gaussian, and I can make this more precise by expanding the exponent around the point