An asymptotic expansion inspired by Ramanujan^{∗}^{†}

Richard P. Brent Computer Sciences Laboratory Australian National University Canberra, ACT 0200

                          rpb@cslab.anu.edu.au

Abstract

Corollary 2, Entry 9, Chapter 4 of Ramanujan’s first notebook claims that

∑_{k=1}^{∞} \frac{(−1)^{k−1}}{nk} (\frac{x^{k}}{k!} )^{n} ∼ ln x + γ

as x → ∞. This is known to be correct for the case n = 1, but incorrect for n ≥ 3. We show that the result is correct for n = 2. We also consider the order of the error term, and discuss a different, correct generalisation of the case n = 1.

1991 Mathematics Subject Classification. Primary 01A60, 33C10, 41A60; Secondary 33-01, 33-03, 33E99, 40A25, 65D20

Key words and phrases. asymptotic expansion, exponential integral, Bessel functions, definite integral, Euler’s constant, harmonic numbers, integrals of Bessel functions, Ramanujan.

1 Introduction

Much of Ramanujan’s work was not published during his lifetime, but was summarised in his Notebooks. These were printed in facsimile in 1957 [14], and edited editions have been published by Berndt [2, 3, 4].

Many of Ramanujan’s results were obtained in a formal manner, and he did not state sufficient conditions for their validity. For example, Example 3, Entry 9, Chapter 4 (page 97 of [2]) is

∑_{k=1}^{∞} \frac{(−1)^{k−1}(φ(k) − φ(−k))}{k} = φ′(0)  (1)

but clearly some conditions on the function φ(z) are required. Berndt has given sufficient conditions, but they do not always hold for Ramanujan’s applications of (1). In fact, Ramanujan does not claim exact equality in (1), but writes that the left side is “nearly” equal to the right side. Thus, some sort of approximation or asymptotic equality is intended.

To illustrate the use of (1), take φ(z) = x^{z}/Γ(z+1), where x > 0 is real. Proceeding formally, we obtain

∑_{k=1}^{∞} \frac{(−1)^{k−1}}{k} \frac{x^{k}}{k!} = ln x + γ ,  (2)

where γ = −Γ′(1) is Euler’s constant.

∗Appeared as Technical Report CMA-MR02-93/SMS-10-93, CMA, ANU, February 1993. Accepted for publication in The Australian Mathematical Society Gazette.

†Copyright \begin{matrix}c \\ ◯ \end{matrix} 1993, R. P. Brent. rpb139tr typeset using LaTEX

If equality in (2) is interpreted as asymptotic equality (usually denoted by “ ∼ ”) as x → ∞, then the result is correct. In fact, a classical result (also given on page 167 of [3]) is

∑_{k=1}^{∞} \frac{(−1)^{k−1}}{k} \frac{x^{k}}{k!} − ln x − γ = ∫_{x}^{∞} \frac{e^{−t}}{t} dt = O(e^{−x}/x) .  (3)

Ramanujan’s Corollary 2, Entry 9, Chapter 4 (page 98 of [2]) is that, for positive integer n,

∑_{k=1}^{∞} \frac{(−1)^{k−1}}{k} (\frac{x^{k}}{k!} )^{n} ∼ n ∑_{k=1}^{∞} \frac{(−1)^{k−1}}{k} \frac{x^{k}}{k!} .

In view of (3), this is equivalent to

∑_{k=1}^{∞} \frac{(−1)^{k−1}}{nk} (\frac{x^{k}}{k!} )^{n} ∼ ln x + γ  (4)

It is plausible that Ramanujan derived this result from (1) in the same formal manner that we derived (2) above, but taking φ(z) = (x^{z}/Γ(z + 1))^{n}.

Berndt [2] shows that (4) is false for n ≥ 3; in fact, the function defined by the left side of (4) changes sign infinitely often, and grows exponentially large as x → ∞. However, Berndt leaves the case n = 2 open.

The aim of this paper is to show that (4) is true in the case n = 2. Theorem 1 in Section 3 gives an exact expression for the error in (4) as an integral involving the Bessel function J_{0}(x), and Corollary 1 deduces an asymptotic expansion. The most significant term is O(x^{−3/2}) as x → ∞.

Theorem 1 is a special case^{1} of a formula given on page 48 of Luke [9]. However, the connection with Ramanujan does not seem to have been noticed before.

In Corollary 2, Entry 2, Chapter 3 of his first Notebook, Ramanujan shows that the function on the left side of (2) can be written as

e^{−x} ∑_{k=0}^{∞} H_{k} \frac{x^{k}}{k!} ,

where H_{k} = ∑_{j=1}^{k} 1/j is a harmonic number (H_{0} = 0). Thus

∑_{k=0}^{∞} H_{k} \frac{x^{k}}{k!} /∑_{k=0}^{∞} \frac{x^{k}}{k!} ∼ ln x + γ .  (5)

In Section 4 we indicate how Ramanujan might have generalised (5) in much the same way that he attempted to generalise (2).

2 Notation and Preliminary Results

In this section we give some preliminary results on integrals involving J_{0}. These results may be found in the literature, but for completeness we sketch their proofs.

Recall that

J_{0}(x) = ∑_{k=0}^{∞} \frac{(−1)^{k}(x/2)^{2k}}{k!k!} .  (6)

J_{0}(z) is an entire function, but we are only concerned with its behaviour on the positive real axis. For small positive x, J_{0}(x) = 1 + O(x^{2}). For large positive x, Hankel’s asymptotic expansion [11, 15] gives J_{0}(x) = O(x^{−1/2}). These observations are sufficient to show that the integral occurring in Lemma 2 below is absolutely convergent.

1Also given in formula 11.1.20, Chapter 11 of Abramowitz and Stegun [1] (the chapter was written by Luke).

In the proof of the following Lemma, (a)_{k} = a(a + 1) ⋯ (a + k − 1) = Γ(a + k)/Γ(a), and

F (a, b; c; z) = ∑_{k=0}^{∞} \frac{(a)_{k}(b)_{k}}{(c)_{k}} \frac{z^{k}}{k!}

is a hypergeometric function.

Lemma 1 For 0 < μ < \frac{3}{2},

∫_{0}^{∞} t^{μ−1}J_{0}(t)dt = \frac{2^{μ−1}Γ(μ/2)}{Γ(1 − μ/2)} .

Proof. The integral^{2} is known as Weber’s infinite integral [16]. We sketch a proof in the case μ < \frac{1}{2}, which is all that is needed below.

Let α be a complex variable. We first evaluate

I(α) = ∫_{0}^{∞} e^{−αt}t^{μ−1}J_{0}(t)dt

for R(α) > 1. Integrating term by term, the power series (6) gives

I(α) = α^{−μ}Γ(μ)F (\frac{μ}{2}, \frac{μ + 1}{2}; 1; −α^{−2}) .  (7)

By analytic continuation, the result (7) holds^{3} in the right half-plane R(α) > 0. Using a wellknown hypergeometric function identity^{4}, we deduce that

I(α) = \frac{Γ(μ)}{Γ(1 − \frac{μ}{2})} (\frac{Γ(\frac{1}{2})F (\frac{μ}{2}, \frac{μ}{2}; \frac{1}{2}; −α^{2})}{Γ(\frac{μ+1}{2})} + \frac{αΓ(−\frac{1}{2})F (\frac{μ+1}{2}, \frac{μ+1}{2}; \frac{3}{2}; −α^{2})}{Γ(\frac{μ}{2})}) .

Since J_{0}(t) = O(t^{−1/2}) as t → ∞, our assumption that μ < \frac{1}{2} makes it is easy to justify changing the order of integration and taking the limit as α → 0+. Thus

∫_{0}^{∞} t^{μ−1}J_{0}(t)dt = lim_{α→0+} I(α) = \frac{Γ(μ)Γ(\frac{1}{2})}{Γ(1 − \frac{μ}{2})Γ(\frac{μ+1}{2})} .

The Lemma now follows from the duplication formula for the Gamma function. □

Lemma 2

∫_{0}^{∞} (\frac{e^{−t/2} − J_{0}(t)}{t}) dt = 0 .  (8)

Proof. A slightly more general result is given in equation 6.622.1 of Gradshteyn and Ryzhik [7], and attributed to Nielsen [10]. We show that (8) follows easily from Lemma 1.

Let μ be a small positive parameter. From Lemma 1,

∫_{0}^{∞}(e^{−t/2} − J_{0}(t))t^{μ−1}dt = 2^{μ}Γ(μ) − \frac{2^{μ−1}Γ(μ/2)}{Γ(1 − μ/2)} .  (9)

Since Γ(1) = 1, Γ′(1) = −γ, and μΓ(μ) = Γ(1 + μ), the right side of (9) is

\frac{2^{μ}}{μ} ((1 − γμ) − (\frac{1 − γμ/2}{1 + γμ/2}) + O(μ^{2})) = O(μ) .

The result follows on letting μ → 0+. □

2More precisely, the generalisation with J_{0}(t) replaced by J_{ν}(t): see Sec. 13.24 of Watson [15].

3A generalisation of (7) is given in Sec. 13.2 of Watson [15], and is attributed to Hankel [8] and Gegenbauer [6].

4See equation (10.16), Chapter 5 of Olver [11].

3 Ramanujan’s Corollary for n = 2

Our main result is the following Theorem, which proves that (4) is valid for n = 2. Theorem 1 Let

e(x) = ∑_{k=1}^{∞} \frac{(−1)^{k−1}}{2k} (\frac{x^{k}}{k!} )^{2} − ln x − γ .

Then, for real positive x,

e(x) = ∫_{2x}^{∞} \frac{J_{0}(t)}{t}dt .

Proof. Proceeding as on page 99 of [2], using the fact^{5} that

γ = ∫_{0}^{1} \frac{1 − e^{−t}}{t} dt − ∫_{1}^{∞} \frac{e^{−t}}{t} dt ,

we have

e(x) = ∫_{0}^{x} \frac{1 − J_{0}(2t)}{t}dt − ∫_{1}^{x} \frac{dt}{t} − ∫_{0}^{1} \frac{1 − e^{−t}}{t} dt + ∫_{1}^{∞} \frac{e^{−t}}{t} dt

= ∫_{0}^{x} \frac{e^{−t} − J_{0}(2t)}{t}dt + ∫_{x}^{∞} \frac{e^{−t}}{t} dt .

Now, from Lemma 2 with a change of variable,

∫_{0}^{x} \frac{e^{−t} − J_{0}(2t)}{t}dt = ∫_{x}^{∞} \frac{J_{0}(2t) − e^{−t}}{t} dt ,

so

e(x) = ∫_{x}^{∞} \frac{J_{0}(2t) − e^{−t}}{t} dt + ∫_{x}^{∞} \frac{e^{−t}}{t} dt = ∫_{x}^{∞} \frac{J_{0}(2t)}{t}dt ,

and Theorem 1 follows by a change of variable. □ Corollary 1 Let e(x) be as in Theorem 1. Then, for large positive x, e(x) has an asymptotic expansion whose leading terms are given by

e(x) = \frac{1}{2π^{1/2}x^{3/2}} (cos (2x + \frac{π}{4} ) + \frac{13 sin (2x + \frac{π}{4} )}{16x} + O(x^{−2})) .

Proof. Using integration by parts and the fact that xJ′′_{0}(x) + J′_{0}(x) + xJ_{0}(x) = 0 (a special case of Bessel’s differential equation), it is easy to deduce from Theorem 1 that

e(x) = \frac{J′_{0}(2x)}{2x} + \frac{J_{0}(2x)}{2x^{2}} − 4 ∫_{2x}^{∞} \frac{J_{0}(t)}{t^{3}}dt .

Continuing in the same way, we obtain an asymptotic expansion

e(x) ∼ \frac{J′_{0}(2x)}{2x} ∑_{k=0}^{∞}(−1)^{k} \frac{k!k!}{x^{2k}} + \frac{J_{0}(2x)}{2x^{2}} ∑_{k=0}^{∞}(−1)^{k} \frac{k!(k + 1)!}{x^{2k}} .

The Corollary follows from Hankel’s asymptotic expansions for J_{0}(z) and J′_{0}(z) = −J_{1}(z). □

5See page 103 of [2].

4 A Different Generalisation

An obvious generalisation of (5) is

∑_{k=0}^{∞} H_{k} (\frac{x^{k}}{k!} )^{n} /∑_{k=0}^{∞} (\frac{x^{k}}{k!} )^{n} ∼ ln x + γ  (10)

as x → ∞.

It is easy to show that (10) is valid for all positive integer n. An essential difference between (4) and (10) is that there is a large amount of cancellation between terms on the left side of (4), but there is no cancellation in the numerator and denominator on the left side of (10). The function (x^{k}/k!)^{n} acts as a smoothing kernel with a peak at k ≃ x − 1/2. Since

H_{k} = ln k + γ + O(1/k) ,

the result (10) is not surprising. What may be surprising is the speed of convergence. Brent and McMillan [5] show that^{6}

∑_{k=0}^{∞} H_{k} (\frac{x^{k}}{k!} )^{n} /∑_{k=0}^{∞} (\frac{x^{k}}{k!} )^{n} = ln x + γ + O(e^{−c_{n}x})  (11)

as x → ∞, where

c_{n} = \{ \begin{matrix}1, & if n = 1; \\ 2n sin^{2}(π/n), & if n ≥ 2. \end{matrix}

In the case n = 2, (11) has error O(e^{−4x}). Brent and McMillan used this case with x ≃ 17, 400 to compute γ to more than 30,000 decimal places. From Corollary 1, the same value of x in (4) would give less than 8-decimal place accuracy^{7}.

The case n = 3 of (11) is interesting because max c_{n} = c_{3} = 4.5. However, no one seems to have used n > 2 in a serious computation of γ.

Although Ramanujan [3, 13] gave many rapidly-convergent series and other good approximations for π, he does not seem to have given series which are particularly useful for approximating γ, except for (3) and (5) above. In his paper [12] on series for γ, he gives several interesting series, of which the simplest^{8} is

γ = 1 − ∑_{k=1}^{∞} \frac{ζ(2k + 1)}{(k + 1)(2k + 1)} ,

but these series all involve the Riemann zeta function or related functions, so they are not very convenient for computational purposes.

Our analysis has assumed that n in (4) and (11) is a positive integer. It would be interesting to consider the behaviour of the functions occurring in these equations for positive but nonintegral values of n, especially in the range 1 < n < 2.

References

[1] M. Abramowitz and I. A. Stegun (editors), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Washington, 1964 (reprinted by Dover, 1965).

6We note an error on page 310 of [5]: in the definition of V_{p}(z), “z/k!” should be “z^{k}/k!”.

7More than 15,000 decimal places would have to be used in the computation to compensate for cancellation of terms Ω_{±}(e^{2x}/x^{2}) in (4) !

8Due to Glaisher: see [12].

[2] B. C. Berndt, Ramanujan’s Notebooks, Part I, Springer-Verlag, New York, 1985.

[3] B. C. Berndt, Ramanujan’s Notebooks, Part II, Springer-Verlag, New York, 1989.

[4] B. C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991.

[5] R. P. Brent and E. M. McMillan, “Some new algorithms for high-precision computation of Euler’s constant”, Mathematics of Computation 34 (1980), 305-312.

[6] L. Gegenbauer, “Über einige bestimmte Integrale (June 17, 1875)”, Wiener Sitzungsberichte 72(2)(1876), 343-354.

[7] I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, fourth edition (trans. Alan Jeffrey), Academic Press, New York, 1965.

[8] H. Hankel, “Bestimmte Integrale mit Cylinderfunctionen”, Math. Ann. 8 (1875), 453-470.

[9] Y. L. Luke, Integrals of Bessel Functions, McGraw-Hill, New York, 1962.

[10] N. Nielsen, Theorie des Integrallogarithmus und verwandter Transzendenten, Teubner, Leipzig, 1906 (reprinted by Chelsea, New York, 1965).

[11] F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974.

[12] S. Ramanujan, “A series for Euler’s constant γ”, Messenger of Mathematics 46 (1917), 73-80 (reprinted in [13]).

[13] S. Ramanujan, Collected Papers of Srinivasa Ramanujan (edited by G. H. Hardy, P. V. Seshu Aiyar and B. M. Wilson), Cambridge University Press, Cambridge, 1927.

[14] S. Ramanujan, Notebooks, two volumes, Tata Institute of Fundamental Research, Bombay, 1957.

[15] G. N. Watson, A Treatise on the Theory of Bessel Functions, second edition, Cambridge University Press, Cambridge, 1944 (reprinted 1966).

[16] H. Weber, “Über einige bestimmte Integrale (Jan. 1868)”, Journal für Math. 69 (1868), 222-237.