therefore
that is;
Putting - for x in the first member of this last equation,
we have
r« zp^ (g — 2 )g-i rf^ __ r( p)r(q)
or r 2^-i(a - 2)^-id« = a^+^-i . -^(^)^(g ) .
j36i?. The last formula in the preceding article is a particu-
lar case of a more general formula by which may be expressed,
in terms of F functions, the multiple integral
fff...xP-hj^-^z''-K..{a-~x — y - z...)'-^dxd2/dz...
extended to all positive values of cc, y, z..., which satisfy the
condition a; + y -|- 2. . . <^ a.
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456 INTEGRAL CALCULUS.
Limiting ourselves to three variables, let
Now, by the last article.
Multiplying this by y^^^dy, and integrating with respect to y
from y==Otoy = a — aj, the result is
(a - xi^+r+.-i nq)nr+B) rir)r(s)
^ ' nq + r + 8)'
and finally, multiplying this last by x^^^dx, and integrating
with respect to x from a? = to a? = a, we have
In this, making a = 1, » = 1, wo have
the limits of integration being any positive values of a?, y, z,
which satisfy the inequality x-^-y ■\'Z <^\.
Assume (^y=„ (tj =„ (t^' =i;
then A = J J J t"'' ¥ 4 '' d^rf^rfj,
subject to the condition that ft + <; + 2; < 1 : therefore
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EULERIAN INTEGRALS. 457
270* By means of Formula 2 of the last article, we can
find the volume bounded by the co-ordinate planes, and the
surface having for its equation
(0"+(f/+e7=
Example. When a = |^ = 7 = 2, and p = g = r = 1, the
surface is that of an ellipsoid of which 2a, 26, 2c, are the axes.
Then, by the formula, the volume V oi \ of this ellipsoid
will be
afte r (2)!
and r (-j = \/^7t; for, let u=:J e-^' dx, then also
J
•'0 •^ "^
Now, / / e"^* ~^* dxdy is obviously \ of the volume the
equation of whose surface is z = e-^*"*'*. In terms of polar
co-ordinates, the expression for the same part of the volume is
r^ r zrdddr= H f e-"-' rdOdr.
But
J e-'^* rdr = - ^e-"* ; .' . T e-"* rdr = ^:
68
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468 INTEGRAL CALCULUS.
and JdOt=d} .-. 2^**^0 = |«;
.*. w = j e~*■*rfa? = HV'''•
Now, r/- j = r e-'x'* dx by defimtion,
— ^f e-^ dif=i2te=^ \/« by patting aBr^sy*:
„ abc \ ^\2)> dbc
therefore, V= -g- —,3 ^A ^ T*
J?7J. Differentiatioti under the sign J enables ns to find
new integrals from known definite integrals. Thus,
dx
n
Example 1. \ —f^ — ='-^a""*.
Difierentiating each member of this equation n times with
respect to a, we get
^» 1.2. 3. ..71 ^ 13 5 271 — 1 1 7t
(a?2 -J- a
)«+! 2 2 2 2 a»-^*2'
/.• c?:t? _ 1.3.5...(2 7t — 1) ;g
whence J ^ ^^, ^ ^^„_^, - 2T476~^^i 2^^^ *
/I
a
After 71 —^ 1 differentiations of the two mdmbers of this
equation, it becomes
g-ax^n-i^-,! 2.3...(7* — l)a^»;
that is J* e-«'aj'»-i cfr = €^ (Art 266),
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DIFFERENTIATION UNDER THE SIGN /. 459
The last formula holds good when a is replaced by the
imaginary quantity a + 6 V — 1, in which a is positive ; for
a+ftV—l
+ G
a + bV^^ +G(Art.73).
therefore
Jo a + bV-l'
and, by differentiating this equality n — 1 times with respect
to a, we get
r . ^../-^, 1^ 1.2.3...(n-l)
j?7;?. The formula just found leads to other integrals by
the separation of real from imaginary quantities.
Assume a + ^V— l=p (cos. 0-^-^—1 sin.^?), in which
Then
•/
= r e"*^(cos.6a; — V — 1 sin. 6a;) a?**"^ da?
J
and
1.2.3...n — 1 _ r{n) 1
(a + 6\/ — 1)** P** cos.n^ + V — Isin.n^
= -~-^ (cos.n^ — \/ — 1 sin. ti^) :
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460 INTEGRAL CALCULUS.
= — ^ (cos.n^ -— \/ — 1 siiL7i(?) ;
an equation which may be separated into the two,
r 1.7. ^(^) .
r(«)
cos.w^.
P'*
j?7$. Making 71 = 1 in the last formula of the preceding
article, it becomes
a^ 4- i^
therefore, denoting by c a constant less than a, we have
ada
But
j^dafy-.o..hxdx^f]^,.
da I e~'^ cos.bxdx = i dx \ e~^ cos.lzda
c •^ J J c
= / cos. toaoj.
J a X
Again :
/a ada _^ 1 a^ + 6^
/ cos.oxax = - I .J , i.» '
Jo ^ 2 c^ + 6^
Making 6 = in the last equation, it becomes
/• e-*'* — e-*" , , a
dxz=l-'j
X c
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J
DIFFERENTIATION UNDER THE SIGN /. 4G1
a result that may also be obtained by multiplying both mem-
bers of the equation
a
by da, and integrating the result between the limits a and c.
274:. In like manner, from the formula
/• b
a^ + 6^
e^*^%ixi.hxdx=.
J
we get
a X 1 c
=z tan.""^ =- — tan.""^ ^ .
6
But
// e^^mi.hxdx^:] dx \ Bm.hxc^da
c •/ J 9 J c
sin. bxdx ;
x
f Bin,iax?a; = tan,~^ =- — tan,~^=-«
J Q X
In this formula, making a = oo , c = 0, it reduces to
/:
-dx = ^
* m,bx 7 «
a? 2
whenJ^;^; whenJ^L.,, the second member becomes — ^:
from which it is seen that the integral / — ^ — dx changes
abruptly from - to — ^, when 6, in passing through zero,
changes from positive to negative.
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462 INTEGRAL CALCULUS,
275. The integral C e-"^ dx=z\y/n (Ex. Art. 270) leads
Jo A
to r e~^dx=z j^Tt; for
J —00 J —00 •/
Now, if we change x into — x, we have
r e-'^'dx^f e-''^dx = ^V^'
f e-"^^ dx z=z j^Ti,
And generally, if /(a?) is a function of the even powers of a?,
that is, such a function that /(x) zzi/^-^ x), then
r /{x)dx=2 r f{x)dx;
for J* /{x)dx=zf f{x)dx+r f{x)dx.
But J f{x)dx=rf{—x)dx=J /{x)dx:
C /{x)dx=2 f^/{x)dx.
In like manner, it may be shown that if/{x) is a function
of the odd powers of x, that is, if /(— x) = — -/(aj), we should
have
r /{x)dxz=0.
e-^ dx=j^7iy putting x^a for
ir, we have
which, by n diflferentiations with respect to a, becomes
J -00 ^ 2*
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DEFINITE INTEGRALS. 463
In this, making a = 1, we have
J —to Ji^
277m Changing a? into ^ + a in the formula
of the preceding article, we get
that is, ^""'/* e-**"2«r^a? = v'^:
But r e-'^-^dx= r c-**-2«;rfa;+ r^e-^^-^^'da?,
J —00 J —00 V
and J e-^^-^'^^cfo^z r"e-**+2aar^a?
by changing a? into — x:
whence
In this equation, replace a by aV— 1 ; then, since
6'^ + e-2-* = e-2ax^^ J. e2a''^~i = 2 cos. 2cca5 (Art. 73),
we have
/* i 1
e""* co&.2axdx=z-e^<^*^/7t.
2 ^
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464 INTEGRAL CALCULUS.
' This example is another instance in which the value of a
definite integral is found by passing from real to imaginary
quantities.
278. Another process by which i e"** cos. 2aoodx may be
found consists in differentiating with respect to a and subse-
quent integration : thus, put
w = I e~'* cos. 2axdx ;
du
then
-— = — I sin. 2aa;e""* 2xd3s = i sin. 2aa?, d, e"*^.
da J J Q
Integrating by parts, and observing^ that, at the limits,
sin. 2aa^""* is zero, we have
— z=z ^f e^'* COS. 2ax2KXjdx — — 2tm:
da J
da
u
But, regarding i^ as a function of a, we have
du
da u da.
Integrating with respect to a, we get
by making e" = C. To determine G, make a = ; then
u = fe-^'dx = l^7t= G:
therefore
/e""'* COS. .200^03 = - f^^j^n.
2
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/e ^0
, —^ = n(c, a, d\
SECTION X.
ELLIPTIC FUNCTIONS.
279. JElUptic Functions or Elliptic Integrals is
the name given to the following integrals : —
— = F(c, 0).
dd^
» (I +asin.^(9) Vl^^^sin:^^ '
The constant c is called the modulus of the fiinction, and is
supposed less than unity ; the constant a, which appears in the
third function, is cflJled the parameter ; and the variable 6 is
called the amplitude of the function. The function is said to
be complete when the limits of the amplitude are and -.
z
The integral of the second order expresses the length of the
arc of an ellipse estimated from the vertex of the conjugate
axis {Art. 240) ; the semi-transverse axis being unity, and the
eccentricity of the ellipse the modulus of the integral. From
this fact, and from the relations whicli exist between the sev-
eral functions, the term elliptic functions has been derived.
Our limits permit us to investigate but a few propositions
relating to such functions.
280. Putting X for sin.t^, the integral of the first order
becomes c' dx
•^0 4/1 — ;
69 466
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466 INTEGRAL CALCULUS.
In like manner, for another value of x denoted by x^j we
have
dxi
•/
Vl-oj'Vl-^c^rcJ
Now assume the relation
^^ + ,. ^^,L„. =0 (1).
Multiply through by the product of the denominators, divide
by 1 — c" a; J , and integrate ; then
J l-c-'x'x' ^J 1-c^xh
1
= constant.
Integrating the first term by parts, we get
J l-c'x'xl '^~ l-c'x^xl
J ' {i-c'x'x\y ■ VT^^,Vi-c*xi
«/ 1 — c^x^x^
In this result, interchanging x and x^ we have the second
term. Adding results, observing that by (1) the terms of the
sum which are under the sign f reduce to zero, we find
a; Vl —xWl- c'^x' + rci Vl - x'' Vl — c'x'
'- ^-^, = const. (2).
Eq. 1 expresses the condition that the variables x and Xi are
so related that the sum of the integrals
dx /•*! dxi
r^ dx /•*! dxi
;
1
shall be constant.
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ELLIPTIC FUNCTIONS. 467
Put r , /i - =T=^ = a, a? = 5^(oc),
Vr^^^ = C7(a), Vl-c^a;^ = i?(a) ;
also
Then, by Eq. 1, we have
* da + d^ = 0:
a -|- ^ = constant = y.
It is also seen from (1) that the constant y is the value of
a?! when a; = ; and, further, when a = 0, we have
therefore, by making the proper substitutions in (2), it be-
comes
^^« + ^)- l-c^j6'(«)}^{^(^)r '
which is the fundamental formula as given by Euler in the
theory of elliptic functions.
281. Suppose the variables <?, di, to be connected by the
equation
r^ do . r<^i ddi p dd
J Vl — c"^8in>^ •^o Vl — c'-'sin.^^i"^'^ o Vl — c^sin.'^^
(1),
or F{c,d) + F{c,d,) = F{c,fx),
in which ^ is a constant. If (?, ^i, be regarded as functions of
a third variable t, and (1) be differentiated with respect to the
latter variable, we have
do do I
Vl -c''8in.2(?^Vl — c2sin.2<?i ^ ^'
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468 IXTEORAL CALCULUS.
Sinco the new variable t is arbitrary, let us assume
do
di
whence, from (2),
1J=V(l-c^8in.'d) (3);
^J = -V(l-c'8in.«<?0 (4).
Squaring (3) and (4), and differentiating, we get
.•. ^?^~372^~ — c'(8m.(?co8.^±sin.(?iC08.^i),
or ^ (<? ifc <?i) = - ^ (sin.2(? ± 8in.2(?i) (5).
Put <? + (?i = qp, and «? — (?i = ip ; then
sin. 2/0 = sin. <)p cos. \ff -\- cos. 9 sin. xpy
sin. 2^1 = sin. <p cos. 1^ — cos. 9 sin. if; :
therefore, from (5), we have
d^cp , . d^xp « .
-jp^ = — c'sin. 9 cos. V, ^r^ = — c* sm. xff cos. 9.
We also have
d^ dip _ /cfe Y _ /^iV — a \ 1 — COS. 20^ 1 — cos. 2/A
dt~dt^\dt) \dt)'^^ ) 2 2 y
„/cos.2^ cos.2<?A . . .
= c^ ( ) = — c^ sin. op sm. yp :
\ 2 2 / ^ r
d^ d^xp
, - =cot.V;, , ^ =cot.9.
agp at/; aqp dip
di di dili
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ELLIPTIC FUNCTIONS, 469
dhp
_ ^ d , . . dcf d jdxp W
But ^^Bm.(y=:cot.(y-^^-, ^^i^- = -^:
dt
d /, d(f\ d d / dxp\ d
— ( 6 — ^ ) = — 6 sm. \p, ~{l—-]=z - l sm. op.
. dt\ dtj dt ^' rf^V rf^/ rf^ ^
Whence
I ^ z=il sin. t/^ + ^> ^ ^ ~ ^ ^^^* ^ + ^i5
or by putting G^IA, Gi = lAij and passing from logarithms
to numbers,
g = ^sin.v^, 5 = ^i8in.g, (6):
. . dip dcp
. • . -d sm. -I/; -7-/ = Ai sm.op ^- ;
dt dt
Acos.\p = Aicos.qi -]- (7).
From Eq. 1 wo see that F{Cjd)=F{c,ii) when <?i = 0:
therefore we then have =:fji= cp =:\pj and (7) then becomes
(A — Ai) COS. fi = 0; and therefore
A COS. (^ — ^1) — Ai COS. (<? + di) = (^ — ^1) COS. fi ;
'whence, by developing cos. (^-—^j), cos. (^ + <?i), and re-
ducing,
{A — Ai) COS. 6 COS. 6i'\-{A-{' Ai) sin. ^ sin. di
= {A - Ai) C03. fi (8).
Now,
t = l + §' = V(l-«"-'»)-V(l-c'™..,,),
-^ = V(l - <^^ sin.2 ^) + y^(l — c2 sin.2 <?i).
Substitute these values in (6), and make 6i=z0; then
\/{l — c^ sin.^ f*) — 1 = ^ sin. ^,
y^(l — c^sin.V) + 1 =-4isin.jM.
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470 INTEGRAL CALCULUS.
From th^se equations, getting the values ofA + Ai,A — Ai,
and substituting in (8), we get, finally,
COS. COS. Oi — sin. sin. O^j^il — c^ sin.^ft) = cos. fi (9).
This relation, by an easy transformation, may be made to
take the form
COS. ^ = cos^iC08.jM -j- sin.^i8in.jMy^(l — c^sin.^^) (10).
Eqs. (9) and (10) express the connection which exists be-
tween the variables in two elliptic functions of the first order
which have a common modulus.
282* Let F{c, 0), F{c, d^), be two elliptic functions in which
c, Cj, and 0, d^ are connected by the equations
2 4c /i\ X /, sin. 2/?i /^.
It is proposed to prove that
Difierentiate Eq. 2, regarding di as the independent variable;
then
1 dd _ 2(1 +c COS. 2^0
cos.'-'^:^ d^i"~ (c + cos.2^i)2
From (2) we also get
cos^^= - (^ + ^0^-2^0^ :
1^2ccos.2^i + c2
dd __ 2(l+cco8. 2^i)
d^ 1 "^ 1 + 2c COS. 2^ 1 -f c2 '
Also, from the same equation, we get
c2sin.2 2^i
1 — c2sin.2(? = l
1 +2ccos.2^i-|-c2
l + 2ccos.2^i + c^cos.^2<?i ,
l + 2ccos.2(?i + c2
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• /
ELLIPTIC FUNCTIONS. ATI
dd
V(l -c^sin.^^)
_ r 2(1 -fc COS. 2^1) y^(l + 2ccos.2^i -f c^) ,
"~J l + 2ccos.2di + c^ l + ccos.2^1 *
+ 2c COS. 2^1 + c^ 1 + c COS. 20 1
ddi
_ 2 r dOj
^'"'Ji-(T^— ■)■
Bnt the last integral, when ■ = Cj , becomes
If we suppose di = -^ 6 = 7t, then
ffc4'i)=^<"""=^4i)
j?55. Having shown (Art. 281) that there exists, between
the variables of two elliptic functions of the first order having
a common modulus, the relation
COS. d COS. di -^ sin. 6 sin. di ^{1 — c^ sin. (a) = cos. i^ (1),
then, between the corresponding functions of the second
order, there exists the relation
£(c, 0) + E{Cf d{) — F{c, fi) = c' sin. sin. ^^ sin. f*.
From the equation between the amplitudes 6, 0^ di, may
be considered as a function of 6 ; that is, we may assume
Eic, d) + E{c, d,) - E{c,(,) =/(<?),
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472 INTEGRAL CALCULUS.
and diflFerentiate, thug getting
By Eq. 10, Art. 281, the first member of this equation may
be put under the form
COS. — COS. di COS. ft COS. 6 1 — cos. d COS. fi ddi
sin. di sin.^ '" sin. d sin. ii dd
_^d{'^m?d -\-%m?di-\' 2cos.(?cos.^iCos.ft 1
do 2sin.^8in.^i8in.^'
But putting Eq. 1 under the form
COS. d COS. di — COS. jM =: ^(1 — c^ siu.^fi) sin*, d sin. di ,
and squaring, we get
cos.^^ -|- cos.^^i + cos.^/i — 2 COS. 6 cos. (?i cos. ^
= (1 — c2sin.V)8in.'^sin.2(?i.
Adding cos.^^j cos.^ft to both sides of this equation, transpos-
ing, and reducing by the relation cos.^ = 1 — sin.^, we find
sin.^ d -j- sin.^ ^i + 2 60s. cos. d^ cos. ^
= 1 + cos.^ II -|- c^^in.' 6 mi? di sin.^f*,
-y- (1 + 008.^^1^ + c^ sin.'^ Bin} 6 1 bvh? fi)
2 sin. <?sin. <?isin.jM
^ . c?(sin. ^sin. ^,)
= cHm.i. ^^— ^:
and therefore, by integration,
/(^d) = c^ sin. ^ sin. di sin. ^
8356
3^
r\
B
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