INDIANA UNIVERSITY JOURNAL OF UNDERGRADUATE RESEARCH
IUJUR Volume II, 201636
ABSTRACT
Construction of Generalized Integral Formulas By Means of
Laplace Transformations
Adam C. Buss, Department of Physics, Indiana University - Northwest
Faculty Mentor: Dr. Axel Schulze-Halberg
generalized integral formula, Laplace transform, definite integral
KEYWORDS:
We present a method for the construction of integral identities that contain an undetermined function. Except for mild
restrictions, this function can be chosen arbitrarily. Our method is illustrated by several examples leading to new integral
identities.
1. INTRODUCTION
T
he closed-form resolution of integrals is one of the
standing issues in calculus. Integrals appear in many
areas of mathematics and in the natural sciences, particularly
in physics and engineering. Existing resources for integral
tables, such as the monographs (Abramowitz & Stegun, 1964;
Brychkov, 2008; Gradshtevn & Ryzhik, 2015) or online data
bases (Shapiro, 2015; Wolfram Alpha, 2016), are continuously
extended as new methods for the resolution of integrals are
discovered. An example for such a new method is presented
in the recent work by Glasser (2013). Starting from a known
identity, the Laplace transform is used to construct the
following integral formula:
(1)
In contrast to the vast majority of identities involving integrals,
(1) contains a function F that can be chosen arbitrarily with
only slight restrictions. As a result, formula (1) can be used to
generate an infinite number of integrals, together with their
closed-form resolution. Consequently, generalized integral
formulas like (1) are much more versatile than their standard
counterparts, such that it is desirable to determine methods
for obtaining them. The purpose of the present research is to
develop an approach for constructing a class of generalized
integral formulas. We start out by considering a group of
functions that have a common class of indefinite integrals.
By imposing condition on the latter indefinite integrals, we
can build integral identities that allow for the construction
of generalized integral formulas (Section 2). As a result, in
Section 3 we obtain several of such formulas, each of which
produces new integral identities.
2. GENERALIZED INTEGRAL FORMULAS
In what follows we will first present an example of a generalized
integral formula and how to obtain it from an identity that
can be found in standard tables. Afterwards, we extend the
example, leading to a scheme of construction for generalized
integral formulas.
2.1. GENERALIZED INTEGRAL FORMULAS
For t > 0 we consider the following identity (Gradshtevn &
Ryzhik, 2014):
(2)
We will now multiply both sides by a continuous function f in
the variable t. Further restrictions on the properties of f will
be developed as our calculation proceeds. After multiplication
by f and reordering terms, we obtain:
(3)
In the next step we integrate this relation with respect to the
variable t over the interval (0, ∞). Furthermore, we require
f to be such that the order of integration can be exchanged.
This leads to the result:
(4)
The terms in curly brackets can be interpreted as Laplace
transforms of the function f. Assuming that f admits such a
transform F, we can rewrite (4) as follows:
(5)
We refer to this identity as a generalized integral formula
because it contains a function F that can be chosen arbitrarily,
as long as it admits an inverse Laplace transform such that the
order of integration in (4) is interchangeable and the integral
exists.
NATURAL SCIENCES
Buss, Generalized Integrals
37
2.2. GENERALIZATION AND METHOD OF
CONSTRUCTION
Before we proceed, let us formulate four standing
assumptions that we will make throughout this work. While
these assumptions will be illustrated using our example in
Section 2.1, they extend to subsequent considerations in a
straightforward manner. First, it is assumed that the function
f, as it appears in the role of a function that is to be integrated in
(3), is continuous. Second, it is assumed that double integrals
that arise during the scheme in (4) allow to change their order
of integration. Third, it is assumed that all integrals converge,
in particular sums or dierences of integrals such as in (4).
Last, it is assumed that the function F contained in the final
result (5) admits an inverse Laplace Transform.
The construction of our integral formula (5) is possible
solely due to a particular structure of our initial identity (2).
In general, the latter identity must be of the form:
(6)
where n is a natural number, A
j
, B
j
, j = 1, ..., n, are functions
and C, D denote constants. However, for several reasons
relation (6) cannot serve as a starting point when it comes
to the practical construction of formulas like (5). This is so
because only very particular choices of functions A
j
, B
j
, j =
1, ..., n, will permit a closed-form resolution of the integral
on the left side of (6). In addition, even if such a resolution
is possible, the integral does not necessarily take the form
shown on the right side of (6). In order to overcome these
issues and construct integral formulas like (5), we will focus
on the function that results from indefinite integration on the
left side of (6).
For the sake of simplicity let us revisit and analyze our example
(2). Indefinite integration gives:
(7)
where Ei stands for the exponential integral (Abramowitz &
Stegun, 1964) and a constant of integration was set to zero.
Next, we must substitute the integration limits. Starting out
with infinity, we make use of the limit relation:
(8)
Applying this to the right side of (7) while taking the limit
gives:
(9)
It remains to evaluate the right side of (7) at x = 0. Since Ei
tends to infinity as its argument approaches zero, we recall
the following series expansion:
(10)
where γ stands for the Euler-Mascheroni constant. On
substituting the arguments of the exponential integral given
in (9), we obtain:
Consequently, we arrive at the expected result:
Let us now consider the following function A that generalizes
the right side of (7):
(11)
where a
1
, a
2
, b and c are functions that are to be determined.
We have:
Now, according to the left side of (6), we want each of the
terms in curly brackets to be a product of an exponential and
a factor that is independent of the variable t. Since we already
have an exponential in both terms, it is reasonable to require:
These conditions are fulfilled if the following choice for b and
c is employed:
(12)
where b
j
and c
j
, j = 1, 2, are functions depending on
a single variable that will be determined further.
Upon substituting (12) into (11), we obtain:
(13)
and:
(14)
INDIANA UNIVERSITY JOURNAL OF UNDERGRADUATE RESEARCH
IUJUR Volume II, 201638
Comparison of this expression with the integrand on the left
side of (6) shows that the functions a
1
, a
2
, can be chosen as
follows:
where k
j
, j = 1, ..., 4, are constants. These settings render (13)
in the form:
(15)
while for its partial derivative (14) we find:
In the next step we will substitute the limits of integration
into (15). Starting out with infinity, we take our identity (8)
into account. This gives the constraints:
(16)
Taking the limit of (15) at zero is more complicated because
the behavior of the functions b
2
and c
2
at zero is not known.
While we are unable to give a general criterion for the limit to
exist, in the example section we will discuss typical scenarios
that lead to a finite limit.
3. APPLICATIONS
We will first take our previous formula (2) and evaluate it
for several particular cases. Afterwards, we construct a new
generalized integral formula (15). In our final example we
show that the latter indefinite integral can also be used to
build formulas involving complex functions.
3.1. EXPONENTIAL AND HYPERBOLIC
INTEGRANDS
We revisit our integral formula (2), which can be obtained
by comparison of the indefinite integrals (7) and (15). Both
coincide if the following settings are employed:
(17)
Let us now state a few particular cases of the generalized
integral formula (5) resulting from (15) with the settings
(17). Recall that we must choose the function F such that
it admits an inverse Laplace transform and such that the
resulting integral exists. Starting out with a simple example,
we plug F(x) = 1/x into (5), which gives:
Another simple example is generated if we choose F(x) = 1/(x
+ 1). We obtain after conversion of exponentials to hyperbolic
functions:
Let us now generate a less elementary integral relation. Upon
setting F(x) =erf(1/x), where erf denotes the error function
(Abramowitz & Stegun, 1964), we get from (5):
Before we conclude this example, let us remark that our
integral formula (2) can be generalized further if we leave
the constants k
j
, j = 1, ...4, arbitrary.
3.2. LOGARITHMIC INTEGRANDS
Let us now make the following settings in (15):
where we further assume that k
1
, k
3
, k
4
> 0. After substitution,
(15) takes the form:
(18)
The partial derivative (14) of this function is given by:
(19)
Keeping this in mind, in the next step we substitute the limits
of integration into the function (18). Starting with infinity, we
make use of (16) in order to obtain:
Next, we observe that the arguments of both exponential
integrals in (18) vanish as x tends to zero, such that (10)
becomes applicable. We get after some simplification:
(20)
Now, combination of (19) and (20) gives the following identity
for t > 0:
(21)
NATURAL SCIENCES
Buss, Generalized Integrals
39
We can now apply the same procedure as in our first example,
that is, multiply (21) by a function f depending on the variable
t, and integrate over the interval (0, ∞). Assuming that f is
such as to allow an exchange of the integration order and
to admit a Laplace transform, we arrive at the generalized
integral formula:
(21)
Recall that this relation holds as long as F has an inverse
Laplace transform. Let us now present a few particular cases
of (22) for dierent choices of the function F. For the sake of
brevity, we first apply the overall settings k
1
= 4, k
3
= 1 and k
4
= 2. Picking F(x) = 1/x renders (22) in the form:
(22)
Next, we plug F(x) = exp(4 − x) into our formula (22). We
obtain:
Note that the right side does not change because we did not
modify the constants k
1
, k
3
and k
4
.
3.3. COMPLEX INTEGRANDS
Except for the introductory formula (1), throughout the
preceding considerations we assumed that the indefinite
integral (15) and the function it contains are real. In
particular, our calculations of the integration limits are
based on real-valued functions. In this section we will show
that choosing complex functions b
2
and c
2
in (15) can lead
to integral formulas that extend those previously studied
here. It is important to point out that our criterion (16) for
determining the limit of A at infinity does not hold anymore
if b
2
and c
2
are complex-valued. While a general analysis of
this case is beyond the scope of this note, we will present
an example. Starting out from the indefinite integral (15),
making the following settings:
This renders our function (15) in the form:
(23)
Its partial derivative (14) becomes:
(24)
We now evaluate the limit of (23) if x tends to infinity. We
obtain:
(25)
Instead of taking zero as the lower limit of integration, this
time we will use negative infinity:
(26)
Upon combining (24) and (25), (26) we get after multiplication
by i/2 the relation:
The form of this identity is suitable for the construction of a
generalized integral formula. Multiplication of both sides by
a function f that has a two-sided Laplace transform F
and exchanging the order of integration yields:
(27)
Before we state particular cases of this identity, let us remark
that due to the complex argument of F and the domain of
integration being the whole real line, the choices for F are
much more restricted than in the previous examples. As
mentioned above, we will not go into details, but just evaluate
(27) for a few cases. The simple setting F(x) = 1 gives the
known relation:
Next, we choose F(x) = x
2
/(x + 1)
2
, which renders (27) in the
form:
Finally, let us substitute F(x) = arctan(x
4
) into (27). We obtain
the result:
As mentioned above, we restrict ourselves to the complex
integral formula (27) because the criteria for choosing the
function F become much more complicated than in the real
case.
INDIANA UNIVERSITY JOURNAL OF UNDERGRADUATE RESEARCH
IUJUR Volume II, 201640
4. CONCLUDING REMARKS
In this note we have presented a simple method for the
construction of generalized integral formulas containing an
almost arbitrary function rather than numerical parameters.
It should be stressed that our method can be generalized by
replacing the Laplace transform through a dierent integral
transform, such as the bilateral Laplace transform or the
Fourier transform. This broadens the choice of functions in
the resulting generalized integral formula and it also allows
to modify the domain of integration.
AUTHOR INFORMATION
All correspondence should be sent to acbuss@iun.edu.
ACKNOWLEDGMENTS
The first author acknowledges funding through the Indiana
University Northwest Undergraduate Research Fund. The
first author would also like to thank Dr. Axel Schulze-Halberg
for his mentorship and support through the various stages of
this research project.
REFERENCES
Abramowitz, M., & Stegun, I. (1964). Handbook of
Mathematical Functions with Formulas, Graphs, and
Mathematical Tables. Mineola, NY: Dover Publications.
Brychkov, Y.A. (2008). Handbook of Special Functions;
Derivatives, Integrals, Series and Other Formulas. Boca
Raton, FL: CRC Press.
Glasser, M.L. (2013). A remarkable property of definite
integrals. Mathematics of Computation, 40(162), 561-
563.
Gradshteyn, I.S., & Ryzhik, I.M (2014). Tables of integrals,
series, and products. In D. Zwillinger & V. Moll (Eds.)
Tables of Integrals, Series, and Products. Waltham, MA:
Academic Press.
Shapiro, B.E. (2015). Table of Integrals. Retrieved from:
http://integral-table.com
Wolfram Alpha (2016). Wolfram Alpha: Computational
Knowledge Engine. Retrieved from: https://www.
wolframalpha.com