Imágenes de páginas
PDF
EPUB

A NOTE ON MEROMORPHIC FUNCTIONS.

By G. N. WaISON.

T has been pointed out by Cauchy that, if

1.

IT

be the Maclaurin expansion of a cothe, valid when <37. then, for all real values of x (whether numerically less than 2w or not), the difference between cothe and the sum of the first + 1 terms of this expansion has the same sign as, and is numerically less than, the (r+2) term. That is to say. for all real values of x,

coth d=c+ c,x2 + ... + c_x* + Oc

where 001.

The proof is very simple. It consists in observing that

[merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][ocr errors][merged small][ocr errors][ocr errors][ocr errors]

it is now obvious that R, and c., have the same sign, and R, is numerically less than c,,,+

* Comptes Rendus, xvii. (1843), p. 374. See also Bromwich, Theory of Infinite Beries, p. 235.

The type of argument just quoted is obviously sufficient to the following theorem, also due to Cauchy:

prove

[merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small]

all positive integral values of r, and if cx is the Maclaurin

m=0

expansion of f(x), valid when x is less than the smallest of the numbers b, then, for all positive values of x,

[merged small][ocr errors][merged small][merged small]

has the same sign as, and is numerically less than, C.,. The reader will have no difficulty in constructing Cauchy's proof.

2. The object of this note is to prove the corresponding theorem for a meromorphic function in which the coefficients of the partial fractions are not all positive, but are alternately positive and negative. This theorem is of some practical importance, as will be seen in § 5. Its proof depends on the well-known lemma that, if u,≥u,≥uu...≥0, then

[ocr errors][merged small]

for all positive integral values of r.

This result is obvious from the fact that the sum may be written in one or other of the forms

(u, − u,) + (u,— u ̧) + ... + (u,_, — u),

[blocks in formation]

[ocr errors]

according as is even or odd; and each of the brackets contains a positive expression.

Stated precisely, the theorem which will be proved is as

follows:

[merged small][ocr errors][ocr errors][merged small]

where 0<b,<b,<b<....., and a„>0 while

*

[ocr errors][ocr errors][merged small]

is convergent. Then, if there exists as integer p such that

[ocr errors]
[ocr errors]

for every value of n, and if cx is the Maclaurin expansion

m=0

of (x) valid when |x|<b1, then, for all positive values of x and for all values of r exceeding p, the equation

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][subsumed][ocr errors][ocr errors][merged small][ocr errors][merged small][merged small][subsumed][subsumed][merged small][merged small][merged small][merged small][merged small][merged small][merged small][subsumed][ocr errors][ocr errors][merged small][merged small][ocr errors][merged small][subsumed][merged small][ocr errors][ocr errors][subsumed]

Since this series is convergent, and since the sequences whose nth terms are b1| (x+b) and 1/ b„ ̃-1 (x> 0, r≥ 1) are monotonic, it follows from Abel's test for

convergence that the series defining (x) and the series £ anb are convergent.

[ocr errors]

It will be proved presently that, if cm be defined thus, then the Maclaurin

[merged small][merged small][ocr errors][merged small]

the first part of the inequality is obvious by the lemma quoted at the beginning of the section; to prove the second part we observe that

[merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][ocr errors]

where 0 ≤0≤1, which is the required result. We observe that, by Abel's lemma,

[ocr errors][subsumed][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small]

as r→ ∞ if 0 < x <b1, so that the series " must con

m=0

verge to the sum (x) when 0 < x <b,, and hence the series. in question is the Maclaurin expansion of (x).

3. It is possible to prove a rather more general theorem than that of § 2, namely that, if

[ocr errors][merged small][merged small]

where Р is any positive number (not necessarily an integer), and a>0, 0<b,<b,<b ̧<..., while 2 (−)"'ab" converges, and if there exists an integer p such that, for every value of n,

n=1

[ocr errors]

then for all positive values of x and all values of r which exceed p

[ocr errors][merged small][merged small][merged small][merged small]
[ocr errors]

where 0 <0≤1, and is the Maclaurin expansion of

m =0

[ocr errors]

(x) valid when \x\<b.

Consider first (x + b)", where b is positive.
Let (a/b) be defined by the equation

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][subsumed][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small]
« AnteriorContinuar »