Mathematical Tools for Physics

1—Basic Stuff 19

Where that last bracketed symbol means “greatest integer less than or equal tox”. It’s a notation more common
in mathematics than in physics. Now in this notation the sum can be written as a Stieljes integral.

∫

f dα=

∫∞

x=0

f d[x] =

∑∞

k=1

f(k) (23)

At every integer, where[x]makes a jump by one, there is a contribution to the Riemann-Stieljes sum, Eq. ( 21 ).
That makes this integral just another way to write the sum over integers. This won’t help you to sum the series,
but it is another way to look at the subject.

Partial Integration
The method of integration by parts works perfectly well here, though I’ll leave the proof to the advanced calculus
texts. If

∫

f dαexists then so does

∫

αdfand

∫

f dα=fα−

∫

αdf

If you have integration by parts, this says that you also have summation by parts! That’s something that you’re
not likely to think of if you restrict yourself to the more elementary notation.

∑∞

k=1

f(k) =

∫

f d[x] =f(x)[x]

∣

∣

∣

∣

∞

0

−

∫∞

x=0

[x]df (24)

If for example the function you’re summing is 1 /knthen

∑∞

k=1

1

kn

=

∫

1

xn

d[x] =

1

xn

[x]

∣

∣

∣

∣

∞

0

−

∫∞

x=0

[x]d(1/xn) = 0 +n

∫∞

1

[x]

xn+1

dx

This particular application isn’t too useful, but there are others that are very useful, particularly in statistical
mechanics.
Still another way to define the integral is called thegauge integral. See for example
http://www.math.vanderbilt.edu/ ̃schectex/ccc/gauge.