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I. F. Quinby Horatio Nelson Robinson.

# A new treatise on the elements of the differential and integral calculus online

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Online LibraryI. F. Quinby Horatio Nelson RobinsonA new treatise on the elements of the differential and integral calculus → online text (page 5 of 22)
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65, Taylor's Theorem may be enunciated as follows:
When a function F{x -\- h) of the algebraic sum of two varia-
bles can be developed into a series arranged according to the
ascending powers of either taken as the leading variable, the
first term is what the function becomes when this variable is
made equal to zero ; the second term is the first power of the
leading variable multiplied by the first diflorential co-efScient
of the first term taken with respect to the other variable ;
the third term is the second power of the leading variable
divided by 1 X 2, and this quotient multiplied by the second
differential co-efficient of the first term ; and tlie {n + 1)'*, or
general term, is the n'* power of the leading variable divided
by the product of the natural numbers from 1 to n inclusive,
and this quotient multiplied by the w'* differential co-efficient
of the first term.

66* In Taylor^s Formula, the co-efficients of the different
powers of the leading variable are functions of the other
variable. When one or more of these functions are such, that,
for a particular value of the second variable, they become in-
finite, the formula fails to give the development of the origi-
nal function for that value of the second variable ; for then
the function ceases to depend on the second variable, and is a

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DIFFERENTIAL CO-EFFICIENTS. 85

function of the first variable alone, and will not necessarily be
infinite for the assigned value of the second variable.
For example, if we have

F{x) = ^,^ZI^^
then F(x + h) = ^(x — a + A).

When rr = a, F{x) = 0, and the first and all the higher
difibrential co-efficients of F(x) become infinite for this partic-
ular value of X ; while, for this value. F{x -\- h) = \/h.

It will be observed that there is a marked difference be-
tween the failing cases for Maclaurin^s and Taylor's Theorems.
When Maclaurin's fails for one value of the variable {x =. 0),
it fails for all ; whereas Taylor's may fail for one value of the
second variable, but give the true development of the function
for all other values.

67m If a function becomes infinite for a finite value of the
variable, its difl'erential co-efficient will be infinite at the same
time. In the case of an algebraic function, this follows from
the fact that such function can become infinite for a finite
value of the variable, only when it is in the forni of a fraction
whose denominator reduces to zero. But the denominator of a
fraction never disappears in the process of differentiation :
hence, if the function has a vanishing denominator, so will its
differential co-efficient. In the case of transcendental func-
tions, it is only by the examination of the different forms that
the truth of this proposition can be established. Thus, in the
logarithmic function y = Zx, y becomes infinite for a? = ;

-J =1 - is also infinite for this value of x ; and for the expo-
ox x

1

nential function y =ia''\ which, if a>l, becomes infinite when

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86 DIFFERENTIAL CALCULUS,

a; = 0, the differential coefficient is -^^ = — - ^a^y 'which is

cue QC

infinite when cc = 0.

The circular functions tan. x, cot. x, sec. x, cosec. x, which
may become infinite for finite values of x, when expressed in
terms of sin. x, cos. x^ are fractional forms to which the reason-
ing in reference to algebraic functions applies.

If a function becomes infinite for an infinite value of the
variable, it does not follow that the differential co-efficient
becomes infinite at the same time.

Thus, in the example y = Zx, ^^ == -, and y is infinite when

dX X

a; = Qo ; but - =r for this value of x,

68. It was remarked in Art. 62, that, unless F{x) and
F{X'-\' h) are such that

i?'(0) + a;Ji"(0) + gj"'(0) + ...,

F{x) + hF'{x) + l'-^F"{x)Jr - .,

give rise to converging series, the formulas of Maclaurin and
Taylor will not serve for the expansion of these functions.

A series in general is a succession of quantities any one of
which is derived, according to a fixed law, from one or more
of those which precede it. If t^07^b^^2?^3j • • • '^n? ^re such
quantities called the terms of the series, then we have

Sn = ^0 + ^1 +^2 + ^3 + • . ''^n-X

for the sum of the first n terms. When this sum approaches
indefinitely a finite limit S, as n continually increases, the
series is said to be converging, and the limit in question is
called the sum of the series ; but, if the sum S„ does not thus

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DIFFERENTIAL CO-EFFICIENTS. 87

approach any fixed limit as n increases indefinitely, the series
is said to be diverging, and has no sum.
The geometrical series

a, ar, ar'^, . . . ar"j

having ar** for its general term, has for its sum

1 — r" fi nr**

" ^ ^ 1— r 1 — r 1 — r

It is evident that, as n increases, this sum converges towards

a

the fixed limit if r is less than 1 ; and that, on the con-

1 — r

trary, as n increases, the sum also increases indefinitely if r is

greater than 1.

We are assured of the convergence of the series

when, as n increases, the sum

Sn = Uo + Ui+U2+ . . . U„_i

converges to a fixed limit Sf and when, at the same time, the
differences

Srt + l — Sn = Un, S„^2 — S„ = U^ + U„^l • • • J

vanish when n is made infinite.

The limits assigned this work do not permit an investiga-
tion of the rules by which, in many cases, the convergence or
divergence of a series may be ascertained.

69. Admitting that F(x) can be expanded into a series
arranged according to the ascending integral powers of Xj
Maclaurin's Theorem may be demonstrated as follows : —

Assume

F{x) = Ao + A,x<' + A,x' + . . . + A,xP
in which A^, A^, A^ ^ , » , Ao not contain x, and the exponents

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88 DIFFERENTIAL CALCULUS,

a,b,c . . . J are written in the order of their magnitude, a being
the least ; then, by successive differentiation, we have
F'{x) = aA^x"-^ + bA.,x^-^ + . . . +pA^x^-\
F''{x)=a{a — l)A^x''-^ + b{b - l)A.,x'-^ + .. .

+ p{p^l)A„xP-\
F'''{x) = a(a — 1) (a — 2)AiX^-' + b{b — 1) (6 — 2) A^x^-^
+ ...+pip-l){p-2)A,xP-\

The assumed and all the following equations, being true for
all values of x, make x = 0; then, since F{0), i^'(O), F''{0) . . . ,
i^puld in general reduce neither to nor to oo, we should have

J, = F{0), a=l, A = F'{0), 6 = 2,

1.2 ' ' ' 1.2.3

F{x)=F{0) + xF'iO) + ^F"iO) + ^F"'{0) + ...,

which is identical with the formula of Art, 62.

70. Xaylor^s Theorem also admits of the following simple
demonstration when the function F(x + h) can be expanded
into a series arranged according to the ascending integral
powers of one of the variables with co-efficients which are
functions of the other variable only.

Assume

Fix + h) =/{x) +Mx) h" +Mx) A» + . . . +/„(x) A",
and differentiate with respect to x, and also with respect to A;
then

dF{x + h) ^df{x) dMx)^^ (y,(x)^, ^/"(^)a'

dx dx dx dx ' " dx

^l^±^\ = oMx)h''-^ + bAix)h''-'+. . . -\-pA(x)b'"K

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DIFFERENTIAL CO-EFFICIENTS, 89

But F{x -\- h) involves h in precisely the same way that it
it does x; and, if we place a; -f 7i = y, we have (Art. 42)

dF{x + h) dF{y) dy _dF{y)

dx dy dx dy

dF{x + h) ^ dF{y) dy^ dFjy)
dh dy dh dy

Xl:

hence dF{x + h) _ dF{x + h) ^
hence ^ _ - ,

that is, these differential coeflScients are equal for all values
of X and h, which can only be the case when they are identi-
cally the same. This requires that

a-\ f(x'\-'ltM. 6-2 /•ra;•>-i-^iM•
also, by making A = in the assumed development, we find

f{x) = F{x);

whence .A {x) = F'{x), /, (x) = ^^ ...:

therefore

Fix + h) = Fix) + hF'ix) + ^F"ix)+...

12

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SECTION VI.

EXPANSION OF FUNCTIONS.

7i. The application of the formulas, demonstrated in tlie
preceding section for the expansion of functions, gives rise
to many important series, some of which we shall now deduce.

1. I{ F{x) = {l+x)"',t\ien

F"{x) =zm(m— 1) (1 + x)""-',

i^('»-i>(x) = m(7n — 1) . . . (m— M + 2)(l -[-a;)"*-""'-^
i^(«) (x) = m (m — 1 ) . . . (/w — 71 + 1) (1 +-a:)"»-'» ;
therefore F{0) = 1. F'{0) = m, F''{0) ^ m (m — 1) . . . ,

^,n-i)(0)=rm (m — 1) ... (m — 71 + 2);
and hence, by Art. 60,

{1 + x)"" = 1 + mx + m -j-2~^' + • • •

+ "* 1.2.3.. .(n-1) "^

When X is less than unity, the last term in this development
will diminish as n increases ; and, by making n sufficiently
great, the series

1 . . rn—l , , (m—l)(m — 2) , ,
Ij^rnx + m -j-^ x- + m ^ j-^^ -' x^ + . . .

90

1
J

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EXPANSTOy OF FUNCTIONS. 91

will approximate more and more nearly the true value of
(1 -|- a:)"* the greater the number of terms taken.

2. Let F{x) — e''; then

F\x) = 6"^ = F" {x) = F^" {x) = .. . = F^""-^^ (x) = i^<"> (x),
F'{0)=:1 = F'' (0) = F'^' (0) ^ , . . = i^(»-i>(0);

X X X

therefore ^' = -^ + T + r2 + TTS "^

^" &r

+ 1.2.3. ..(n-l)"''l.2. ..»*
Making in this x^l, we have

a series that may be used for .finding the approximate value
of e.

3. Let F(x) = sin.rc; then

F^ {x)=: cos.a;= sin. f a; + -),
rfsin. [oj + oj

^"<»^)= — W^=»'-(-+l)=""-('+¥}

n

Therefore F{0) = 0, F' (0) = 1, F" (0) = 0, F'" (0) = - 1,
and we have

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92 DIFFERENTIAL CALCULUS,

x^ x^

a;"-^ . n— 1

+ i72:3:TT(;r:riy'°- -2- "

+ 172737:7 n^'^-l^^^+T/

4. Let -f (x) = COS. X ; then

i?''(a;) = — sin. x = cos.fx + ^ j, jP" (x) = co8.f a; + — - j,

F'" (x) = COS. fx + ~\... F'"^ (x) = COS. (x + ^\
i^(0) = 1, i?" (0) = 0, i?"' (0) = - 1, F'" (0) = . • .
i?'<"-i>(0) = cos.^^ff:

k ^ X'^ X*'

hence cos.xzir 1 — j^+ ^ 0.3.4

H COS. 7t

^1.2.3. ..(/I- 1) 2

4- :r- COS. to -^ ).

^ 1.2.6. ,.n \ ^ 2/
By Art. 63, it will be observed that the last terms in Exs.
2, 3, and 4, diminish as n is increased, and finally vanish when
n becomes infioite.

5. Let i^(a;) = f(l+x); then

F(0) = 0, F' (x) = 1, F" {x) = — \, F'" (x) = 2 . . .
i?'"' (0) = ( — 1)»- > 1.2.3. . . (n - 1) ; and therefore
7/1 . \ x' x' X* , (-1)"-* „ ,

(-1)"-^ _x^

+ » (I + to)"'

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EXPANSION OF FUNCTIONS. . 93

An examination of the last term of this expansion shows,
that^ when x does not exceed unity, this term necessarily de-
creases as n increases, and vanishes wlien n becomes infinite.

And, since the factor [ -r-r-ir ) under this hypothesis cannot
exceed unity, the sum of the series, up to the n^^ term in-
clusive, cannot differ from the true sum by more than - • and

, hence, by increasing n sufficiently, this difference can be made
as small as we please.

Changing the sign of a;, we have

7/1 N ^' ^' ( - l)«-2 „ ,

n (X — Oxf
6. Let F{x)z=zi^xir^x; then

1-1- a;- 2Vl-a;V-l ^l + a;\/-l
= I ((1 - xV^)-i + (1 + xy/^Vr
F"(x)=.\ X ^ 1 X - \/=l (1 - x-oZ—iy
+ ^ X - 1 X V=l ( 1 + a; V^^ )-'

= V:

- 1 X ^((l - ^ V- ir - (- 1)^ (1 + a;^/^!)-* j

F"'{x)

1.2

= ( V- 1)' y ((1 - xV^^)-' - (- 1)' (1 + x V^I)-')

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94 DIFFERENTIAL CALCULUS.

therefore

i^c. (0) = (v/^T)"-' kM^^(!L-_L) U - (- 1)").

Whence it follows, that, if n is an even number, i^<'"(0) = 0;
but, if n is uneven, then

i?'<»> (0) = (v/=ri)»-i L?:!i_-^(?i:zl) X 2

= i^-lZ 1.2.3.. (7i-l) = ± -7-''^^-1.2.3..(n-l).

Hence we have

, x^ ^ x^ x^~^

tan.~* x^=zx rr + -r- • • • ± -i

3 5 71 — 1

^ (1— ^^V^)~"^ (1 + rV^)"" ^

^"^ 2V-T

The final term in this development is not in a convenient
form, as it stands, to decide whether the series is converging
or diverging ; but by referring to Ex. 18, p. G7, making

a = 1, and observing that there ^ = ^ — tan.~^ x, we have

F^n) (^) ^ (_ 1) n-1 1.2.3. ..(n-l) ^.^^ fn^ _ ^ ^^^ _i \ .

{i + x'f \2 y

therefore
tan "^ a:i=:iC— — +-r & + •••

o o

(-ir^^ — ^sin.f^;-ntan.-^a:

This form of the final term shows, that, if x is less than
unity, the numerical value of the term may be made as small
as we please by giving to n a value sufficiently great.

The above form for i^^"^ {x) might have been used for find-
ing all the differential co-efficients of tan.~^ x as readily as that
specially deduced for that purpose.

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EXPANSION OF FUNCTIONS. 95

The following is a more simple process for getting the
expansion of tan.~^ x : —

Assume

tan.-^ x=:A + Bx+Cx'- + Dx^ + &c. (1 ),
and differentiate both members with respect to x ; then

j^^ = B + 2Cx + 3J)x-^ + &c. (2);

but by division, or by the Binomial Theorem,

, } , =:zl-x'' + x'- X^ + X^ -.& + &C. (3).

The second members of (2) and (3) must be identical: hence,
equating the co-efficients of like powers of x, we have

o

and, since the assumed development must be true for all values
of ic, make a; = in (1), and we find A = 0: therefore

tan -1 0? ^ X — —+ - - - + &

7. If y = sin.~^x, assume

sin -^x = A + Bx+ Cx^ + Dx"" + . . . (1),
and differentiate both members ; then

^ ^ = B + 2Cx-\'ZDx''-\-4.Ex^ + . . . (2);

yf X ~-~ X

but, by the Binomial Theorem, we find

1 „ . 1.3 . . 1.3.5

The co-efficients of the like powers of x in the second
members of (2) and (3) must be equal: hence

and, by making ic = in (1), we get ^ = : therefore

. _i , li»' , 1.3 a;^ , 1.3.5 a:^ ,
Bin,~* X:=.X -+• V- f- . . .

^2 3 ^2.4 5 ^2.4.6 7 ^

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96

DIFFERENTIAL CALCULUS:

8. \jei y =z e' ""■ *, and assume
y=A, + A^xJrA^x' - \-.. . + J„x" + . . . (1).
Differentiate twice ; then

^-f =Ai-\-2A^x + ZAiX^ + ... + nA^x''-^-lr. . . (2)

= 2Jj + 2.3J,x + ... + («_ l)n^„a;— * + . . .• (3).
But

dy _ a.in-^x «

dx Vl-a;«

and hence

— e -|- , ;

Substitute in (4) the values of -^y - 4, taken from (2) and

dx dx^

(3), and we have

2A2+2.SA3X + SAA.x^-] h (^ - l)n^„a?""' + . .

'-'{2A2X^' + 2.SA,x^+SAA,a^-\ \'{n — l)nA„x'' + ..)

— ( AiX + 2A^x^ + 3A^x^ + 4:A^x* -] f-^^«^'*+ • •) .

Equating the co-efBcients of the same powers of x in the
two members of this equation, we find

T^" "^'-~2.T

and generally

-<»2 — "5" -^c -^» o"? — -^1' ""* — g~J~-^«**''

^. = «_!±(!L^A_....(5).

(n — l)n

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EXPANSION OF FUNCTIONS, 97

If then ^0 and ^1 be found, formula 5 will give all the
following co-efficients in terms of these two.

Aq is what e"**'"" "^ becomes when x=zO: hence ^q = 1.

And ^1 is what ^r^e"**^-"^^ — r=^ l^^comes whena7 = 0:
dx Vl-x'

hence Ai==^a.

In formula 5, making n equal to 2, 3, 4, &c., successively,
we get

^ _ «' ^ -i^!+A)^ A - («' + ^ ')«'

^""O' '~ 1.2.3 ' ^*- 1.2.3.4 •••

Substituting these values ofu4«, AifA^. . . in the assumed
development, it becomes

«.,«-u 1 , , a^x'- , a(a2+l) . , aVa^ + 2') ,

+ 172^75 ^■+-"

By Ex. 2 of this article, avo also have
c'""'"''=l + asin.-ix-h~(sin.-ia;)^ + ,-^(sin.-'a;)3 + ..

Equating the co-efficient of the first power of a in this
series with that of the same power of a in the preceding
series, we have

. , , 1 a;3 1.3 a;^ , 1.3.5 X' , ,

^2 S^2A 5 ^2.4.6 7 ^ '

as in Ex. 7. By equating the co-efficients of a^, we should

also find

/ • -1 v> , 22 ^ , 2^4'-^ 6 , 2"^. 41 62. 3
(sm. ^-)- = -+o" +3.47570" +3.4X(>:T:8^-"

9. V = Hl+en^ y=^^+-2+h-2^ik3A+^

13

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98 DIFFERENTIAL CALCULUS.

11. 2^ = Z(l + sin.a;), y nz a; — ^4-^ . . .

12. y = e'"- ', y = i + x + ^_^-^|...

13. If y = ( — J __-w — ^ 1 , sliow for what values of x

Taylor's Theorem fails to give the development.
It fails for a; = c; 1st term is then infinite.
It fails for x=:a; 2d differential co-efficient is then infinite.

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i

SECTION YIT.

APPLICATI05J OP SOME OP THE PRECEDING SERIES TO TRIGO-
NOMETRICAL AND LOGARITHMIC EXPRESSIONS.

72. Let a and b represent any two real quantities what-
ever; then a-^bs^— 1 will bo the most general symbol for
quantity, since, by giving to a and b suitable values, it may be
made to embrace every conceivable quantity, real or imaginary.

The two expressions, a + b V— I, a — 6 V— 1, which dif-
fer only in the signs of their second terms, are said to be conju-
gate ; and their product, (a + 6 V — 1 ) (a — 6 V — f ) = a^ -[- 6^,
is always real and positive. The numerical value of the square
root of a^ 4" ^^ IS the modulus of either of the conjugate ex-
pressions. Denote this modulus by r; then it may be shown
that the expression a-\-b\^— 1 can be put under the form

r{cos.O + V- Isin. 6).
For let a 1= r cos. Ojb^=zr sin. :

tan./? = -, r2(cos.2/? + sin.2/9)iz=r2=:a2-fi2,

r = \/a''-\-b\

7t

^ Now, if we suppose the arc of a circle to start from— -, and

^, to increase by continuous degrees to -f- o? passing through

t^ zero, the tangent will at the same time increase by continuous
degrees, and pass through all possible values between — oo
and -|- 30 . Among tliese values of the tangent, there must be

one that will satisfy the eauatky^an. ^ = - ; and the arc an-

478155. ^ ,

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100 DIFFERENTIAL CALCULUS,

Bwering to this tangent Avill be that whose sine and cosine will
satisfy the equations a = r cos. Oj b=zr sin. 6, and therefore
render r{co&.0 + V— I sin. 6?) the equivalent of a-|-6 V — 1.
73. Let us resume the series (Art. 71, Exs. 2, 3, 4).

cos.x = l - +^^^^-...(3)
and in (1) write x V— 1, — x V— 1, for x successively ; then

, ; x^ x^\/—\ ^*

-i-t-a;v i ^^ ^2.3 ^^ 1.2.3.4

"^1~2"3"4~5'' == cos.a; + V-lsin.a;,

as is seen by comparing this result with the second members
of(2)and(3).

x'^ x^

Also e-^^-^ = 1- xV- 1 - Ti; +

V^

1.2 ' 1.2.3. • 1.2.3.4

COS. X — V — 1 sm. 03 :

1.2.3.4.5 '
therefore cos. x + V— 1 sin. x r= e^^^-^ • • • (4)

cos.ic — V— 1 sin.a;=: g-^^-i . . . (5),
also COS. y + V— 1 sin. y = e^^-^ • • . (6);

multiplying (4) by (6)
(cos. 0? + V— 1 sin. x) (cos. y + V — 1 sin. y) = e(^+y)^-i

= cos. (iz^ + y) + V— 1 sin. (x + 2/)-

Eflfecting the multiplication in the first member, and then

equating the real part in one member with the real part in the

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TRIGONOMETRICAL EXPRESSIONS, 101

other, and the imaginary part in the one with the imaginary

part in the other, wo find

COS. [x-\-y) =■ COS. X COS. y — sin. x sin. y

sin. (x -\-y) =z sin. x cos. y -[- sin. y cos. x.
Again :

(cos.a:-f-V — Isin.ir) ^cos.?/4-V — Isin-y) (cos.z + V— Isin.^) • •
= e^^+^+'- • >"^-i= COS. (x+y-\-z + ' •)+ V^sin.(aj+y+2+. ^),
from Avhich, by making x =^y =iz= • • • , wo have

(cos. X + V — 1 sin. ir) "* zzr cos. twj; -{- V — i sin. mXj
and generally

(cos. a; ± V — 1 sin. x) "* = cos. mx i V — 1 sin. mXj
which is known as De Moivro'.s Formula.

Hence the multiplication of expressions of the form of
COS. X -\- \^ — i sin. Xj and therefore of all imaginary expres-
sions, is thus reduced to an addition, and the raising to
powers to a multiplication.

7^:. Dividing formula (4) of the preceding article by (5)
of the same, member by member, we have
e"^^"^ _ 2^v— I __ ^'^^'^ + V— 1 sin X __1 -\- V— 1 tan. a?.

€~^^~^ cos. a; — \/ — Isln. a; 1 — V— 1 tan. a:

whence, by taking the Napierian logarithms of both members,

2xV^^ = I (1 + V^i tan. x) —I (I— V^^l tan. x) .

Expanding the terms in the second member by Ex. 5, Art. 71,

/ — - / — - , tan.^x / — tan.^ic tan.'* a;

2xV— 1 = V— Itan.irH ry V — 1

/ — ^-tan.^ic

2 ^ " 3 4

x
5~

-f- y_-I tan. .V + -'— + V^n.
tan."*.:; , tui.'^.j

tan.' X

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102 DIFFERENTIAL CALCULUS.

Equating tlio imaginary parts in the two members of this
equation, and then dividing through by 2 V— 1, wo have

tan.^ir , tan.^a; i\\n? x , . .
X = tan. X — H > — _ f- & — <fcc.,

o O I

a series that may be used for the calculation of tZj and which*
agrees with the formula in Ex. 6, Art. 71.

7 J. To find the expansion of cos.'*aj in terms of the cosines
of multiples of re.

Make e'^^-'^^y; tlien e''^^~~^=y'^j e-'"^^ = ",

^_w-^~i___

y
1

From formulas 4, 5, Art. 73, Ave find

2 cos. X = 6^""^! + c-'^-^ = 2/ + ~

2 V — 1 sin. X = e-^^-^ — - '""

y

1

^ y

also, from De Moivre's Theorem, we deduce

2 COS. mx = i/"' -}- - ^7 2 V — 1 sin. mx = y''* — -^.
»/ y

1 / l\n

Because 2 cos. a; = y + - , 2" cos." a; = ( y + - ) 5
but

. n-1 1 , 1,1

•••+'' i.2~r^^ + ''p^+r'

by combining terms at equal distances from the extremes;
hence

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TRIGONOMETRICAL EXPRESSIONS, 103

COS. "a; = -K^zi ( ^^s. 7ix-\-n cos. {n — 2)x

Since there are n-}-l terms in series (a), when n is even, the
number of terms is odd, and the middle term, that is,

n(n-l)(n-2)...Q+2)0+l)

,2 / 2

will be independent of y, and consequently of x ; but, when n
is odd, n + 1 is even, and there is no middle term in series (a),
and therefore no term independent of x. In the first case,
there will be within the ( ) in formula (6), besides the term

that does not depend on x, ~ terms, containing as factors the

first COS. no:, the second cos. (n — 2) a:; and so on to the last,
which will have cos. 2x for a factor. In the second case, that
is, when n is odd, there is no term within the ( ) in formula

(6) that does not involve x; but the — ' — terms will then have

Ex. 1. cos.'*a; = -^ ( cos. 4a; + 4 cos. 2x -[- 6

for factors, severally, cos. nx, cos. (n — 2)x, . . , cos. Zx, cos. x.
23

Ex. 2. cos.^a; = — f cos. 5a; + 5 cos. 3x + 10 cos.x ].

2^\ /

76. To find the expansion of sin.^cc in terms of the sines
of multiples of x.

By formulas 4 and 5 of Art. 73, we have, employing the
notation of the last* article,

y

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(«)•

104 DIFFERENTIAL CALCULUS.

therefore 2''(V— 1)' 9in.»a; = 2''(— 1)^ 8in."a; = (y — -]

^ \

= y» - «y»-« + n -y-^- y"-* - &-\

An examination of tliis series shows, that, when n is even,
the second terms within the ( ) are all plus ; and, when n is
odd, they are all minus. In the first case, the expansion of
^ui.^x will involve only the cosines of multiples of x ; and, in
the second case, it Avill involve only the sines of these multi-
ples.

n

The factor (—1)^ in the first member will be positive and
real when n is any one of the alternate even numbers begin-
ning with 0; that is, when n is or 4 or 8 or 12, &c. ; and
negative and real when n is one of the alternate even num-

n

bers beginning with 2. In like manner, (—1)'^ will be imagi-
nary and positive when n is any one of the alternate odd
numbers beginning with 1 ; and it will be imaginary and nega-
tive when n is any other odd number.

Let h represent any positive whole number, zero included ]
then the different series of values above indicated for n will be
embraced in the four fonns, 44, 44 + 2 ; 44 + 1, 4fc + 3.

It would be of no advantage to make formula (a) conform
to each of these cases by special notation, as it can be easily
applied, as it now stands, to the examples falling under it.

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TRIGONOMETRICAL EXPRESSIONS. 105

Ex. 1. Expand sin.' a? in terms of the sines of the mul-
tiples of X,

2'(V^)'sin.»x = ^y» _ 1) _ 3 ^y - ^)

= 2V-^ 1 sin. Sx — 6V— 1 sin. a?;

1

sin.'ic = — — (sin. 3a; — 3 sin. x).

Ex. 2. Expand sin.*aj in terms of the cosines of the mul-
tiples of x.

= 2 cos. 4a; — 8 cos. 2a; -|- 12 :

sin.^a; = — f COS. 4a; — 4 COS. 2x + 6 ].

Ex. 3. Expand sin.^o; iu terms of the sines of the mul-
tiples of a;.

= 2 V^^ sin. 5a; — 10 V^^ sin. 3a; -f 20 V^sin. a?:

sin.^ic ^^ o4 ( ®'^* ^^ — ^ ^^^* ^^ "f" ^^ ^^^' ^ )•

Ex. 4. Expand sin.^o? in terms of the cosines of the mul-
tiples of X.

2e( V:ri)esin.ea;z= ^ye+ 1) - 6 {y^^ 1) + 15 (y^+l) - 20

= 2 cos. QiX — 12 COS. 4a; + 30 cos. 2a; — 20 :

8in.*a;=: — — (cos. 6a;— 6 cos. 4a; + 15 cos. 2a; — 10).
2'\ /

77. To find the different n*^ roots of unity.
Let X represent the general value of the n*^ root of unity;
then, by the definition of the root of a number, a;" = l, or

14

(

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106 DIFFERENTIAL CALCULUS.

a;" — 1 = 0; and the object of the investigation is to find all of
the values of a: that will satisfy the equation x^ — 1 = 0.
By Do Moivre's Theorem, Art. 73, we have

(cos. y dt= V— i sin. ?/)'" = cos. my dt= V— 1 sin. my;
an equation which holds, whether m is entire or fractional,
positive or negative. Now, if h be any whole number, 2k7t

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