CHAPTER 1

The Gamma function extended to

nonpositive integer points

This first chapter discusses how to extend the Gamma function to nonpositive

integer points and serves as a preparation for the more general issue as how to

make sense of certain types of divergent integrals. Whereas here the divergence

is at zero, later in the notes the divergences will take place at infinity, but the

way one cures these divergences is similar. The Gamma function offers a good toy

model to compare regularisation methods mentioned in the Preface. Extending the

Gamma function to nonpositive integers arises as an instance of the more general

problem of extending homogeneous distributions at negative integers. We show

that Riesz and Hadamard’s “finite part” regularisation methods lead to the same

extended homogeneous distributions (see Theorem 1.24) and hence to the same

extended Gamma function, a feature which arises again later in these notes. We

discuss discrepancies induced by the regularisation procedure, which are a first hint

to further obstructions we will encounter while working with regularised integrals.

1.1. Homogeneous distributions

Let

S(R+)

= {f ∈

C∞(R+),

∀(α, β) ∈ Z≥0 × Z≥0,

∃Cα,β, s.t.

|xα∂βf(x)|

≤ Cα,β ∀x ∈

R+}

denote the space of Schwartz functions on

R+

:= ]0, +∞[. The following ex-

ercise shows that Schwartz functions on

R+

are smooth functions f on

R+

whose

derivatives

∂βf(x)

go faster to zero as x tends to infinity than any inverse power

x−α.

Exercise 1.1. Let f ∈

C∞(R+).

Show that

f ∈

S(R+)

⇐⇒ lim

x→+∞

(

xα∂βf(x)

)

= 0 ∀(α, β) ∈ Z≥0 × Z≥0.

Decreasing exponentials are typical Schwartz functions.

Example 1.2. Show that the map φ : x →

e−x

defines a Schwartz function on

R+.

One can build homogeneous distributions given by linear forms on

S(R+)

in

the following manner.

Exercise 1.3. Given a Schwartz function φ on

R+

and any complex number a

with real part Re(a) larger than −1, show that the map x →

xa

φ(x) lies in

L1(R+).

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http://dx.doi.org/10.1090/ulect/059/01