1 dx -1/2. k 2s x 2r. ( 2s )!! -1/2-1/2. k 2r x 2r

Transcription

1 5 Highe and Supe Calculus of Elliptic Integal 5. Double seies epansion of Elliptic Integal 5.. Double seies epansion of Elliptic Integal of the st kind Fomula 5.. The following epessions hold fo Poof F(,k ) 0 Since k,. d - -k (-) + -s s k + (.) ( -s- )!! ( s- )!! k s + + ( -s )!! ( s )!! (.') k,, fom Genealized Binomial Theoem (See 3. ) k - Multiplying each othe, it is as follows k - That is k k k + k k k k 0 + k 0 k 0 + k + k k - -s s k s - -s s Then, integating both sides of this with espect to fom 0 to, we obtain (.). Net, fom the definition of Geneal Binomial Coefficient (See 3. ), k s - -

2 s ( +) ( -s+) ( s+) ( /) ( /-s) ( s+) On the othe hand, fom the popeties of the Gamma Function (See..6 ), n + Then, fo non-negative intege s, Using these, i.e. s n -!!, n, -s (-) s s (-) s ( s- )!! (-) s s s! (-) s ( s )!! s-!!, -s Hence, substituting these fo (.), we obtain (.'). Eample : Double seies epansion of F, -n (-) n s ( s- )!! ( s- )!! s s! n ( n -)!! (-) -s ( -s- )!! ( -s )!! When abitay point 0.7 is given to this elliptic integal and its double seies, both ae compaed and the ight side is illustated, it is as follows. - -

3 5.. Double seies epansion of Elliptic Integal of the nd kind Fomula 5.. The following epessions hold fo k,. Poof E(,k ) 0 Since -k - d - -s s + ( + )( -s) / ( -s- )!! ( -s )!! k,, fom Genealized Binomial Theoem k - / - k k s + (.) ( s- )!! k s + (.') ( s )!! Multiplying each othe and integating both sides of the esult with espect to fom 0 to, we obtain (.). This is only what evesed the sign of / in the nd binomial coefficient in (.). Net, fom the definition of Geneal Binomial Coefficient, / s ( 3/) ( 3/-s) ( s+) ( 3/) ( /-s) ( /-s) ( s+) On the othe hand, fom the popeties of the Gamma Function (See..6 ), the following epessions hold fo non-negative intege s, 3 Using these, / s (-) s ( -s) s! Substituting this and pevious, -s (-) s -s Eample : Double seies epansion of E, s ( s- )!! ( s- )!! (-) s ( s- )!! s ( -s) ( s )!! ( -) -s ( -s- )!! fo (.), we obtain (.'). ( -s )!! When abitay point 0.8 is given to this elliptic integal and its double seies and both ae compaed, it is as follows

4 5..3 Tiple seies epansion of Elliptic Integal of the 3d kind Fomula 5..3 The following epessions hold fo Poof (,c,k ) 0 Since c, k,. d +c - -k s t0 + t s t0 - c -s s-t (-c ) -s ( s-t- )!! + ( s-t )!! c, k,, fom Genealized Binomial Theoem. +c (-) c - - -k k k t + (.3) ( t- )!! k t + (.3') ( t )!! Multiplying each othe and integating both sides of the esult with espect to fom 0 to, we obtain (.3). And eplacing the binomial coefficients with the double factoial in (.3), we obtain (.3'). Eample3 : Tiple seies epansion of,, 3 When abitay point 0.9 is given to this elliptic integal and its tiple seies and both ae compaed, it is as follows

5 5. Ac length of an ellipse The ellipse which has foci on -ais is shown by the following fomula. y a + b ( 0<b <a ) () Compaing this with cos + sin, i.e. a cos, y b sin acos, y bsin, 0 This is illustated as shown in Fig.A. By this epession, since the ac length is calculated by the counteclockwise otation fom -ais, the calculation is complicated. Theefoe we think about only the st quadant, and let us eplace with /-. i.e. asin, y bcos, 0/ (') Then, we can calculate the ac length clockwise fom y-ais as shown in Fig.B. Because the calculation is easy, we adopt epession ('). Fig.A Fig.B When the paametic equation of a cuve on a plane is f( ), y g( ), the length l is given by the following epession. Fom ('), l d d dy + d Then d dy acos, -bsin d d d d dy + d a cos + b sin a - a -b sin a - k sin (k a -b /a : Eccenticity ) - 5 -

6 Theefoe, the length l of bq is as follows. l a 0 - k sin d (.0) Futhemoe, let t sin ( /a ). Then, a l a 0 -k t -t dt ae a,k k a a -b (.) Since the ight side is a Elliptic Integal of the nd kind, using (.) in pevious section, we obtain l a - -s s + / + k s a k a -b a (.') Eample : The length of bq in the following figue Fom a 3, b, q, k 3-3 q, 35 a 3 Then the length l of bq is as follows fom (.'). l 3 - -s s + When this is calculated, it is as follows. / + 35 s 3-6 -

7 5.3 Ac length of a lemniscate Definition of a lemniscate The locus of the point fo which the poduct of the distance fom some fied points on a plane is a constant is called Lemniscate of a wide sense. Especially, when the numbe of the fied points is, and the distance is FF' c, and the constant poduct is c, it is called Lemniscate in a naow sense. The equation is dawn fom FP F'P c 4 and is as follows. Othogonal coodinates : + y - c - y 0 Pola coodinates : -c cos 0 Although these equations ae used with this, the following fomula eplaced by c a is often used. In this chapte, these ae used heeafte. Othogonal coodinates : + y - a - y 0 (3.0o) Pola coodinates : -a cos 0 ( a >0 ) (3.0p) Eplicit function (Othogonal coodinates) Fom (3.0o), y Since a > 0, y - +a a 8 +a These zeos ae obtained fom 0 a, 0 8 +a > 0, Employing +, we obtain - +a a 8 +a (3.0h) -a 0 which is dawn fom (3.0h) 0. They ae as follows. That is, the -coodinate of A points is a in Fig.. The epession of + of (3.h) is an uppe half (blue) and the epession of - is a lowe half (ed) in Fig.. Relation between othogonal coodinates and pola coodinates Substituting a cos fo cos, y sin, we obtain the following equations. a cos cos, y a cos sin (3.y) - 7 -

8 Net, substituting the following epessions fo (3.y), cos -sin, cos -sin we can etact sin as follows. sin 3a a a +8 a Futhemoe, fom this, we obtain. a a a -8y (3.0') a sin - 3a a a +8 sin - a a a -8y (3.0") a a Ac length fom the ight end point (pola coodinates) In Fig., the figue of each quadant is symmetical with a point, and also symmetical with a line. So, we pay ou attention only to the st quadant. Then, the length l of AP in Fig. is obtained as follows. Diffeentiating (3.y) with espect to, d -a sin d acos sin cos - cos dy a cos d asin sin cos - cos Fom these, Hence, d d l a 0 dy + d d cos a cos a 0 d -sin 0 4 (3.0) 5.3. Elliptic integal epession of ac length (Pat ) Let t sin in (3.0), then dt cos d, sin t, cos - t - 8 -

9 Fom these d - sin - t dt cos dt -t - t Then, the length l of AP in Fig. is obtained as follows. l a u dt 0 -t - t a F u, (3.) u sin 3 a +8 - a (3.u) Whee, (3.u) is obtained fom t sin and (3.0'). Since the ight side of (3.) is the elliptic Integal of the st kind, using (.) in pevious section, we obtain l a - -s s + s u + (3.') Eample: Ac length fom a to 0.8 at the time of a (Pat ) Note k F is given by the following epession. When K L 4,k is a complete elliptic integal of the st kind, the length L/4 of AO in Fig. a K a (3.q) Futhemoe, the length l of OP in Fig. is dawn fom (3.) and (3.q) as follows. a l K - F t, ( 0t ) (3.) 5.3. Elliptic integal epession of ac length (Pat ) du Let t tan in (3.0), then d +t fom tan - t. On the othe hand, - 9 -

10 cos + sin -sin cos -sin +tan +t -tan -t Then, the length l of AP in Fig. is That is, l a 0 u l a 0 u +t dt -t +t dt -t -i t a 0 u dt -t +t. af( t,i ) (3.) u tansin - 3a a a +8 (3.u) a Whee, (3.u) is obtained fom u tan and (3.0"). Although the ight side of (3.) is slightly iegula ( k - ), since it is the elliptic Integal of the st kind, using (.) in pevious section, we obtain the following epession. l a - -s s + i s u + Eample: Ac length fom a to 0.8 at the time of a (Pat ) (3.') Single seies epession of ac length In the pevious two eamples, elliptic integal was calculated by the double seies. On the othe hand, (3.) is calculable by single seies. Accoding to the Genealized Binomial Theoem, Fom this, o - 4 l a 0 u l a 0 u dt -t 4 dt -t 4 - a a ( - )!! ( )!! u The uppe limit of the integation is the same as (3.u), as follows. u tansin - 3a a a +8 a u 4+ (3.3) (3.3') - 0 -

11 Eample3: Ac length fom a to 0.8 at the time of a (Pat 3) Using (3.3'), we ty to calculate Eample by hand. The significant figue is to thee numbes below a decimal point. The uppe limit of (3.3') at the time of a and 0.8 is same as Eample. Then, u tansin ( 0.8) Substituting this fo (3.3') and calculating the fist 5 tems, we obtain the following. (-)!! s 0!! !! +!! !! + 5 4!! !! + 9 6!! !! + 3 8!! Note Afte all, thee ae the following elations between the thee integals. d t dt 0 0 -sin -t - t 0 u du -u 4 t sin sintan - u u tan tan sin - t u +u t -t - -

12 5.4 Temwise Highe Calculus of Elliptic Integal 5.4. Temwise Highe Integal of Elliptic Integal Fomula 5.4. When, k, c F(,k ) 0 (,c,k ) 0 d - -k d +c - -k the following epessions hold fo natual numbe n. Poof 0 F(,k) d n 0 0 E(,k) d n 0 0 (,c,k) d n 0 Fom Fomula 5.., F(,k ) + (-) ( )! ( +n + )! (-) ( )! ( +n + )! s t0, E(,k ) 0 -s s / -s s s-t t (-) ( )! -s c ( +n + )! - -s s - ( )! ( + )! -s s k s + k s + -k - d k s +n+ (4.) k s +n+ (4.) k t +n+ Integating both sides of this with espect to fom 0 to n times, we obtain (4.). (4.) and (4.3) ae also obtained fom the Fomula 5.. and 5..3 in a simila way. Note The following epession using the double factoial is also possible. Howeve, it is complicated and does not have a meit. 0 F(,k) d n ( )! ( -s -)!! ( s -)!! 0 k s +n+ ( +n + )! ( -s)!! ( s )!! Eample : 0 0, d 3 3 In the left side, the integand of elliptic integal of the nd kind is integated 3+ times by Riemann-Liouville integal. In the ight side, this highe integal is calculated by the double seies (4.). Moeove, one abitay point 0.8 is given to the both sides, and the values of the both sides ae compaed. Both ae coesponding. (4.3) - -

13 5.4. Temwise Highe Deivative of Elliptic Integal Fomula 5.4. When, k, c, F(,k ) 0 (,c,k ) 0 d - -k d +c - -k the following epessions hold fo natual numbe n. Poof d n d n F(,k ) n - d n d n E(,k ) n - d n d n (,c,k ) Fom Fomula 5.., F(,k ) n - (-) ( )! ( -n + )! (-) ( )! ( -n + )! s t0 (-) ( )! ( + )!, E(,k ) 0 -s s (-) ( )! -s c ( -n + )! / -s s -s s -k - d k s - n+ (4.4) k s -n+ (4.5) s -t t k s + k t -n+ Diffeentiating both sides of this with espect to n times, we obtain (4.4). The st tem 0 of outside is decided as the powe -n + of is nonnegative. And thee is a following elation between the st tem 0 and the ode n of deivative. n (4.6) Such a elation can be epessed as follows using a ceiling function n - 0 n 0. (4.5) and (4.6) ae also obtained fom the Fomula 5.. and 5..3 in a simila way

14 Eample : d 3 d 3,, 3 In the left side, the integand of the elliptic integal of the 3d kind is diffeentiated 3- times diectly. In the ight side, this highe deivative is calculated by the double seies (4.6). Moeove, one abitay point -0.9 is given to the both sides, and the values of the both sides ae compaed. Both ae coesponding

15 5.5 Temwise Supe Calculus of Elliptic Integal 5.5. Temwise Supe Integal of Elliptic Integal Fomula 5.5. When, k, c F(,k ) 0 (,c,k ) 0 d - -k d the following epessions hold fo p 0. 0 F(,k) d p 0 0 E(,k) d p 0 +c - -k 0 (,c,k) d p 0 (-) ( )! ( +p +) (-) ( )! ( +p +) s t0, E(,k ) 0 -s s / -s s (-) ( )! -s c ( +p +) -k - d k s +p+ (5.) k s +p+ (5.) s -t t k t +p+ Poof Analytically continuing the inde of the integation opeato in Fomula 5.4. to [ 0,p ] fom[,n ], we obtain the desied epessions. Eample : 0 0 F, d 3 In the left side, the integand of elliptic integal of the st kind is integated 3/+ times by Riemann-Liouville integal. In the ight side, this highe integal is calculated by the double seies (5.). Moeove, one abitay point 0.7 is given to the both sides, and the values of the both sides ae compaed. Both ae coesponding, (5.3) 5.5. Temwise Supe Deivative of Elliptic Integal Fomula 5.5. When, k, c, - 5 -

16 F(,k ) 0 (,c,k ) 0 d - -k d the following epessions hold fo p 0. d p d p F(,k ) +c - -k p - d p d p E(,k ) p - d p d p (,c,k ) p - (-) ( )! ( -p +) (-) ( )! ( -p +) s t0, E(,k ) 0 -s s (-) ( )! -s c ( -n +) / -s s -k - d k s -p+ (5.4) k s -p+ (5.5) s -t t k t -p+ Poof Analytically continuing the inde of the diffeentiation opeato in Fomula 5.4. to [ 0,p ] fom [,n ], we obtain the desied epessions. Eample : d 3/ E, d 3/ 3 In the left side, the nd ode deivative of elliptic integal of the nd kind is integated / times by Riemann- Liouville integal. In the ight side, this supe deivative is calculated by the double seies (5.5). Moeove, one abitay point 0.6 is given to the both sides, and the values of the both sides ae compaed. Both sides ae coesponding, (5.6) Alien's Mathematics K. Kono - 6 -