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Now by a known theorem, when n is even, we have

sin nz

sin ≈

- 2-1 sin (2) sin(+2)...sin(x);

=

which, when ≈ = 0, gives

2

(n 2)

n2− (n − 1) = (sin)" (sin 2)'... {sin (")}",

so that we find

n

2n

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2nd.

:

When n is odd, there is no middle factor, and the number of double factors is (n - 1). Also in this case we have

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and by making = 0, we have as before

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2n

n-1

π

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but for the demonstration the reader is referred to Legendre's work.

To this definite integral (n) others may be easily reduced.

Thus if we have the integral

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fo da aTM-1 (log )***,

by assuming = x, it becomes

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on differentiating 2n times with respect to a and then making a = 1, we obtain

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according to the notation adopted by Euler, and, after him by Legendre; the value of the integral being supposed to change in consequence of the variation of p and q, n remaining constant. This form of the integral however is not the most convenient in practice, and we shall use another, formed from the present by putting a=y, when it becomes

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Putting P

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=

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n

n

= M, we shall designate the definite

integral by a symbol of functionality applied to these letters as the symbol I is used for the second Eulerian integral: the letter we shall use is the digamma F, so that we write

(a)

a'

fo da a-1 (1-x)m-1 = F (l, m).

The most important properties of this integral are those by which it is connected with the second Eulerian integral founded on the theorem

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To demonstrate this we shall proceed as in Ex. (1, e), which is a particular case of this theorem;

г (1). г' (m) = f f dx dy e-(+) x2-1 ym−1;

and on transformation

г (1). г (m) = fo du e-" u'+m-1 f1 dv v1-1 (1 − v)TM−1.

Now

Lo du e-"u'+m-1 = F(l+m) and fo' dv v'-1 (1-v)m-1 = F(l,m) ;

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As and m enter symmetrically into the second side of the equation, it follows that

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and as the second side is a symmetrical function of l, m, n, it follows that these letters may be interchanged, so that

(d) F(l,m).F(l+m, n) = F (l,n). F (l+n,m)=F(m, n).F(m+n,l).

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the limits of y being the same as those of x. Hence

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(3) The property (b) of the first Eulerian integral may be extended to a large class of multiple integrals by the following theorem due to M. Lejeune Dirichlet*.

(a) Let Vfdx fdy fdz..."-y-1-1...

in which the limiting values of x, y, z... are given by the condition

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a, b, c...a, ß, y...p, q, r...being positive quantities;

I'

a

r

then will "..."()()()...

V

=

a

pqr...

T (1+

a b

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+...)

The equation of the limits may be made linear by putting

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Hence if we know the integral

U = fdx fdy fdx... -1 ym-1 x-1...,

with the previous condition for determining the limits, we can find I.

When the variables are two in number, it is easy to see that the integral is identical with that called the first Eulerian. Let us suppose therefore that there are three variables. Then Ufdæ a dyy"- fidzz"-1,

1

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Assume = v≈1, yuy1, the limits of u and v are then O and 1, and U takes the form

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But as y1 = 1-x, and x, y, uy, = (1 − x) (1 − u), the integral becomes

U = f' dx x'-1 (1 − x)”+”. fo' du uTM-1 (1 − u)”. fo1 dv v”−1.

The integrations with respect to the different variables may now be effected separately, and we have

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In like manner we might find the value of U when there are four variables, and so on for any number; and hence, also, the value of V, as stated, is deduced.

(b) By a similar process M. Liouville has proved the still more general theorem, that if

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-1

P

y

W=fdx [dy fdx..." ~`y'-'*'-'...ƒ {()2‍+()' + (-)-+--}·

where the limits are given by the condition

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