Mathematical Tools for Physics

2—Infinite Series 40

Differentiate 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

]

+···

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) (12)

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

)

(13)

The binomial series, Eq. ( 3 ), for the case of a positive integer exponent is

(1 +x)m=

∑m

n=0

(

m

n

)

xn, or more symmetrically

(a+b)m=

∑m

n=0

(

m

n

)

anbm−n (14)

(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 choosenobjects from a
collection ofmof them,mCnis the answer.