It is rather easy to find integrals that cannot be expressed in “closed form”, which in this context means functions whose properties we know a lot about. I, for one, am rather grateful for the resources — and the people who compiled them! — such as the Table of Integrals, Series, and Products, DLMF, AMS55 and Wolfram Math World that we may consult for calculus results and properties of “special functions”.

In this post I will discuss how to evaluate the following two integrals by solving the relevant differential equations they satisfy.

(I):

(II):

Here, is the Bessel function of the first kind; is the modified Bessel function of the first kind; and

(III): .

**Motivation** Since there are infinitely many intractable integrals anyway, you may wonder why you should pay attention to this result. There is in fact a physical reason for doing so. I hope to start writing about it more, but in curved spacetimes, waves associated with *massless* particles in Nature — including light itself — do not in fact travel strictly on the null cone. Results (I) and (II) describe the inside-the-light cone (aka “tail”) portion of *massive* scalar waves in de Sitter spacetime, by viewing the latter as a hyperboloid situated in 1 higher dimensional Minkowski spacetime. The massless wave tails can in turn be obtained by setting .

**Derivation of I and II** By a direct calculation, you may readily verify that

(D1):

where here is either or . Note that this *is* the ordinary differential equation (ODE) satisfied by itself; i.e., .

In more detail, you should find that applying upon the left hand sides of equations (I) and (II) yields

(D2):

and

(D3):

That is, acting on converts the integrands into total derivatives, which then tells us the result is simply these integrands evaluated at the end points . But from eq. (III) we see these are precisely the zeroes of and hence of and ; which in turn means the integrals are zero. In other words, . But as already alluded to, this is precisely the ODE satisfied by itself. However, since there are two linearly independent solutions, we still need to show that our integrals satisfy . For non-integer , note that are linearly independent. Moreover, we may check from equations (I) and (II) that are in fact power series in that begin with an overall pre-factor arising from the . Since is times a positive power series in , this tells us there cannot be a term in our . What remains is to figure out the in

(D4):

To do so, notice cannot depend on . We may therefore extract their values through the limits

(D5):

From equations (I) and (II), and utilizing the Taylor series results

and

we have

(D6):

and

(D7):

From equations (D4), (D5) and (D7), we may see that equations (II) has been proven for non-integer . What remains, therefore, is to tackle eq. (D6). If we put

(D8):

so that

eq. (D6) now reads as

(D9):

Since the cosine is an even function we may extend the integration limit to , namely

(D9′):

At this point, referring to the integral representation of the Legendre function — see here, for instance — tells us we have arrived at the right-hand-side of eq. (I), at least for non-integer . Our results are very likely true for integer as well, since I believe it is safe to assume they are continuous functions of .

References

- Y.-Z. Chu, “
*A line source in Minkowski for the de Sitter spacetime scalar Green’s function: massive case*,” Class. Quant. Grav.**32**, no. 13, 135008 (2015); doi:10.1088/0264-9381/32/13/135008; [arXiv:1310.2939 [gr-qc]].