This historic book may have numerous typos and missing text.
Purchasers can download a free scanned copy of the original book
(without typos) from the publisher. Not indexed. Not illustrated.
1903 Excerpt: ... a: . 2xz, 2.4a;5, 2.4.6X1. _ 4/1-2 3 3-5 3-5-7
From which, multiplying both members by 1--x2, we obtain 15.
(i--jr) ' sin-1. = x H R DV ' 335357 in which, putting x = sin 6,
we have sin20 2 sin40 2 4 sin60, 16. 0 cot 0 = 1 h It. 3 3 5 3 5 7
When the determination of the successive derivatives of a higher
order is laborious, a simpler method may be employed provided the
development off'(x) is known. Thus, since sin-1 x is an odd
function, which vanishes with x, we assume f(x) = sin-1. = Ax + Bx
+ G5-j-Dx1 + etc. Differentiating, we have /'() =, = A + zBx + $Cx
+ iDx + etc. (1) /i--x Developing 1/ 4/1--x, we have, provided x 1,
V ' 2 ' 2.4 ' 2.4. 6 ' 3.5.7...(-l)+ ( ' 2.. 4. 6... 2ft W or,
placing Ir1 731 r X 6=j5" = A' = ' 30 = i?" etC-'-v v /v /y--v v-8
The coefficients represented by 2?, 2?3, 2?s, etc., are used in the
higher branches of analysis, and are called Bernoulli's numbers,
128. Extension of Taylor's Formula to functions of two or more sums
of two variables each. Let u=f(x, y), x and jy being independent
variables; and let it be required to develop /(x--/iy y--k), in
which h and k are variable increments of x and respectively. If,
in/(#, y), x be increased by h, and/(#--h9y) be developed by
Taylor's formula; then if, in each term of the result, y be
increased by k and developed in a similar manner as a function of y
+ k, we shall have/(# + k, y + k) = sum of the latter developments.
Otherwise, develop f(t) = f(x + ht, y+ kt) as a function of t, by
Stirling's formula, and in the result make = 1. 0W = /( + M + ),
.-0(0) = /(, j) =. In order to express conveniently the successive
deriva tives with respect to /, place x + ht = ze/, and j +--giving
0(/) = /(-/, J). Hence ( 102), dcp(t) _dP(t)dw df(t)ds dt dw di '
ds df Su...
|Country of origin:
Edgar W. Bass
||246 x 189 x 3mm (L x W x T)
||Paperback - Trade
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