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which is a general formula, of which (336) is the particular case, when m = 1. Also in (339), let m = 5: then, since r(t) = at,

1.3.5 ... (2n-1) = 2*+n" e-", when n = cc. (340) 138.] Many other curious and important properties of the Gamma-function may be deduced from the preceding equations, and will be found in monographs on Eulerian Integrals, and in treatises where these properties are specially investigated. They scarcely fall within the narrower scope to which this treatise is limited: and I must refer the student to other works for them. I purpose to conclude this section with the application of the Gamma-function to the evaluation of some important definite integrals, and to the theory of series.

Let us first consider the more general form of the Gammafunction ; viz. le-az".vn–1 dx.

In this integral let – ax" be replaced by -X; then


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and as the value of this last integral is not changed when x is replaced by -X,

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If in these integrals b is replaced by b N-1, we have

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Next let us take the more general form of the Beta-function

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which is indeed the same result as (17), Art. 82. Hence also, if r = 2n,

po 2-1 dx

do 1+ ax2u – 2na wbich result may be proved by common integration.

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and | e-a*x*-1 sin bx dx, when n is a positive quantity greater than unity, also depend immediately on the Gamma-function.

Let c = 16-ax xn-1 cos bx dx ;

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C+8V-1=; (cosna + ✓-1 sin na);

(a? +62); (cos so that equating possible and impossible parts, and replacing c and s by their values, I e-az gn-1 cos bx dr – f(n) cos na .

(348) (a2 +62)

r(n) sin na e-4220"-sin bx dx =


(a? +62) which results have already been found by Cauchy's method in Art. 104.

It will be observed that in the preceding process of evaluation the exponent of e contains an impossible as well as a possible part, and it may consequently be supposed that the method is not rigorously exact. Much might be said on the subject; let it however suffice to say that as the possible part of the exponent is negative, and as only positive values of x are included within the range of integration, the results are doubtless correct. They may also be verified in the following way.

Expanding cos bx we have

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r(n) cos na

(a+62) if b = a tan a. Similarly may (349) be verified.

140.] If in the preceding expressions a=0, then a = z; and (348) and (349) become, when n is a positive quantity,

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Hence also, replacing r(n) by its value given in (278),

NTT 2-1 cos bx dx = in

bar(1-n) sin nt


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1-n being a positive proper fraction. Let 1-n be replaced by m, when m is a positive proper fraction;

pocos bx dx 1m-1 m . Jozo = 2r(m) sec 2 ;

(355) posin bx dx

Tom-1 Jo zysha = 2r(mj cosecom. (356) Hence if m = 1, which is its superior value, po sin bx ,

(357) Jo which is the value already determined in (76), Art. 94.

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141.] Also from these formulæ two other integrals of considerable importance may be deduced. By reason of (250), Art. 122, e-a-xv=1)2 2-1 dz =

(a-~ -]'

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(a? + x2)n la and dividing both members by xm, and then making them the element-functions of an x-integral between the limits oc and 0, we have

poo (a +XN – 1)" dx | dx pas r(n) % scontato czy?"* = 5 e-(0-2v=1820-1 dz

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