Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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2.1 DIFFERENTIATION


The pattern emerging is clear and strongly suggests that the results generalise to


f(n)=

∑n

r=0

n!
r!(n−r)!

u(r)v(n−r)=

∑n

r=0

nC
ru

(r)v(n−r), (2.14)

where the fractionn!/[r!(n−r)!] is identified with the binomial coefficientnCr


(see chapter 1). Toprovethat this is so, we use the method of induction as follows.


Assume that (2.14) is valid fornequal to some integerN.Then

f(N+1)=

∑N

r=0

NCrd
dx

(
u(r)v(N−r)

)

=

∑N

r=0

NC
r[u

(r)v(N−r+1)+u(r+1)v(N−r)]

=

∑N

s=0

NCsu(s)v(N+1−s)+

N∑+1

s=1

NCs− 1 u(s)v(N+1−s),

where we have substituted summation indexsforrin the first summation, and


forr+ 1 in the second. Now, from our earlier discussion of binomial coefficients,


equation (1.51), we have


NC
s+

NC
s− 1 =

N+1C
s

and so, after separating out the first term of the first summation and the last


term of the second, obtain


f(N+1)=NC 0 u(0)v(N+1)+

∑N

s=1

N+1C
su

(s)v(N+1−s)+NC
Nu

(N+1)v(0).

ButNC 0 =1=N+1C 0 andNCN=1=N+1CN+1, and so we may write


f(N+1)=N+1C 0 u(0)v(N+1)+

∑N

s=1

N+1C
su

(s)v(N+1−s)+N+1C
N+1u

(N+1)v(0)

=

N∑+1

s=0

N+1C
su

(s)v(N+1−s).

This is just (2.14) withnset equal toN+ 1. Thus, assuming the validity of (2.14)


forn=Nimplies its validity forn=N+ 1. However, whenn=1equation


(2.14) is simply the product rule, and this we have already proved directly. These


results taken together establish the validity of (2.14) for allnand prove Leibnitz’


theorem.

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