Classical Borel and Laplace transforms

In this section we present some elementary properties of the Laplace transform and of its formal inverse, the Borel transform. For an in depth treatment the reader should refer to one of the numerous books on the subject, we mention [Sch99] as an example. The Laplace transform is usually defined as an integral from 0 to ∞. Here we will use a slightly more general definition.

Definition 2.1. Let θ ∈ R and ˆφ be such that r 7→ φˆ(reiθ) is analytic on a

neighbourhood of R+ and |φˆ(s)|6Ceτ|s|. Functions with this property are called functions of exponential type along the direction θ. If the previous bound holds in every direction, we just say that the function is of exponential type. Moreover if τ0

is the infimum of all such τ we say that the function is of exponential typeτ0. We define the Laplace transform in the direction θas the linear operator Lθ,

Lθ_{φˆ(t) :=}

Z eiθ∞

0

e−tsφˆ(s)ds.

The function Lθ_φˆ_{is analytic on the half-plane Re (te}iθ_{)> τ}0_{, see Figure 2.1.}

We define the convolution of two functions by

f∗g(s) =

Z s

0

θ

1

θ₂

Figure 2.2: If ˆφis analytic and of exponential type τ in a sector, then the domain of analyticity of L(θ1,θ2)_φˆ_{is the union of all possible half-planes.}

The Laplace transform transforms convolution to multiplication. i.e.

Lθ_{[ ˆf∗fˆ] =L}θ_{[ ˆf]·L}θ_{[ ˆf].}

If the function ˆφis analytic in a sector{s∈_C|θ1 <args < θ2}, with θ2−θ1 < π, and is of exponential typeτ in that sector, then the Laplace transform converges on anyθin the sector and the functionLθ1_φˆ_{is the analytic continuation of}Lθ2_φˆ_{. So}

we can define the functionL(θ1,θ2)_φˆ_{which is analytic in the union of the domains}

of analyticity of Lθφˆfor all θ in the sector, see Figure 2.2. We can define L(θ1,θ2)_φˆ_{also when π < θ}

2−θ1 <2π. The situation is essentially the same with the only difference that the functionL(θ1,θ2)_φˆ_{might be multivalued.}

Let θ ∈ (0,π₂). If ˆφ is of exponential type τ in the sectors Sθ = {s ∈ C| −θ <

args < θ} and S−θ ={s ∈ C|π−θ < args < π+θ} but it is not of exponential

type in C\(Sθ)∩S−θ, then one can define L(−θ,θ)φˆ and L(π−θ,π+θ)φˆ. See Figure

2.3. However at the points t where both are defined their difference cannot be identically equal to 0.

If ˆφhas a pole at finite distance from the origin, then its Taylor series has a positive but finite radius of convergence. This implies that the Laplace transform of its Taylor series, applied termwise, has 0 radius of convergence.

Let C[[s]] denote the space of formal power series of s with complex coefficients.

We denote byt−1

Figure 2.3: The domains ofL(−θ,θ)φˆand L(π−θ,π+θ)φˆ.

constant term.

Because R₀∞s_nn_!e−tsds=t−n−1 for Ret >0, we have for any θ

Lθ sn n! (t) =t−n−1, Re (teiθ)>0.

Using this we define the formal Laplace transform Lθ :C[[s]]→t−1C[[t−1]].

Definition 2.2. The formal Borel transform is the linear operator

B: φe(t) = X n>0 cn tn+1 ∈t −1 C[[t−1]] 7→ φˆ(s) = X n>0 cn sn n! ∈C[[s]].

Notice that the Borel transform is formally the inverse of the Laplace transform. This means that since the Laplace transform turns convolution into multiplication, the Borel transform turns multiplication into convolution.

If φe has a positive radius of convergence, if for example it converges for t−1 < ρ,

then ˆφdefines an entire function of exponential type ρ−1.

Let φe∈t−1C[[t−1]] be divergent and let ˆφ=Bφe∈C[[s]] have a positive radius of

convergence. Still it may happen that ˆφextends analytically and is of exponential type in sectors. In these cases the Laplace transform converges in each sector but generally each sector defines a different function. This implies that we can possibly get a function from a formal series by taking the Laplace transform of the analytic continuation of the Borel transform of the series.

We denote by E the analytic continuation of a function defined as a convergent power series around the origin. We define

I(θ1,θ2)₌L(θ1,θ2)_{◦ E ◦}B

to be the Borel Laplace summation over the sector (θ1, θ2). Similarly if for some ˜

φ ∈t−1C[[t−1]], I(θ1,θ2)[ ˜φ] is a function analytic at a domain like the one in right

figure of 2.2, we say that ˜φis Borel-Laplace summable.

The Borel Laplace summation is regular, i.e. it sends a convergent series to its function. It is linear and it commutes with multiplication, differentiation, integra- tion, translation of the argument and composition. These imply that if ˜φ satisfies formally some analytic equation then its Borel Laplace sum satisfies the same equa- tion.