The Volterra functions 
and 
are defined for 
, real 
, and real 
by the integrals (Erdélyi
et al. 1955, pp. 217-218)
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
is the gamma function and reciprocals of 
are taken to vanish at its poles.
In particular, 
.
For 
and real 
, the Laplace
transform with respect to 
is (Garrappa and Mainardi 2016)
 |
(3)
|
The transform follows by interchanging the integrals and evaluating the inner integral in 
with the gamma function.
In particular, the Laplace transform of 
is 
. These functions arise in solving Volterra
integral equations of the first kind with a logarithmic integral
kernel (Garrappa and Mainardi 2016).
The special case 
can be written as the difference between the exponential
function and an integral due to Ramanujan (Garrappa
and Mainardi 2016),
![nu(t)=e^t-int_0^infty(e^(-rt))/(r[pi^2+(lnr)^2])dr.](/images/equations/VolterraFunctions/NumberedEquation2.svg) |
(4)
|
The Volterra functions will be implemented in a future version of the Wolfram Language as VolterraMu[beta, alpha, t] for 
and VolterraNu[alpha,
t] for 
. The shorter forms VolterraMu[beta,
t] and VolterraNu[t] set 
.
