完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z,...
TRANSCRIPT
![Page 1: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/1.jpg)
完全楕円積分と楕円関数
![Page 2: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/2.jpg)
( ) ( )( ) [ ) [ )
( ) ( ) ( )( ) [ )
∈∈−=
∈∈−−
= ∫
1,0,,0,,sn1:,cn
1,0,1,0,11
:,sn
2
0 222
1-
kkKzkzkz
kzk
dkzz
ξξξ
( )kz,sn ( )kz,cn
( ) ( )cn , sechk →・ ・Jacobi’s sn function
Jacobi’s elliptic functions sn(z,k) and cn(z,k):
![Page 3: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/3.jpg)
( ) ( )( ) [ ) [ )
( ) ( ) ( )( ) [ )
∈∈−=
∈∈−−
= ∫
1,0,,0,,sn1:,cn
1,0,1,0,11
:,sn
2
0 222
1-
kkKzkzkz
kzk
dkzz
ξξξ
( )kz,sn ( )kz,cn
1),(sec),( →→ kaskzhkzcn
![Page 4: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/4.jpg)
( ) ( )( ) [ ) [ )
( ) ( ) ( )( ) [ )
∈∈−=
∈∈−−
= ∫
1,0,,0,,sn1:,cn
1,0,1,0,11
:,sn
2
0 222
1-
kkKzkzkz
kzk
dkzz
ξξ
ξJacobi’s elliptic functions sn(z,k) and cn(z,k):
1/4 period of ).,(),,( kzcnkzsn
( )( )( )
12
2 20 02 2 2:
1 sin1 1
d dK kkk
πξ φ
φξ ξ= =
−− −∫ ∫
Complete elliptic integrals of the first kind
![Page 5: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/5.jpg)
Notations: complete elliptic integrals
∈−+
=Π
∈−=
∈−
=
∫
∫
∫
20 222
20
22
20 22
)1,0[ ,sin1)sin1(
:),(
)1,0[ ,sin1:)(
)1,0[ ,sin1
:)(
π
π
π
ϕϕν
ϕν
ϕϕ
ϕ
ϕ
kk
dk
kdkkE
kkdkK
( ) ( )( ) [ ] [ )∫ ∈∈−−
=z
kzk
dkz0 222
1- 1,0, 1,0,11
:,snξξ
ξJacobi’s sn function
![Page 6: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/6.jpg)
( ) ∫ −= 2
0 22 sin1:
π
ϕ
ϕ
k
dkK ( ) ∫ −= 2
0
22 sin1:π
ϕϕ dkkE
( )( )
( ) ( ) 1as01,
20
2 ↑→−
∞→
=
kkKkkK
K π( )
( ) 112
0
=
=
E
E π
![Page 7: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/7.jpg)
について ),( kνΠ
∫−+
=Π 20 222 sin1)sin1(
:),(π
ϕϕνϕν
kdk
5.0=ν のとき
![Page 8: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/8.jpg)
![Page 9: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/9.jpg)
![Page 10: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/10.jpg)
![Page 11: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/11.jpg)
![Page 12: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/12.jpg)
![Page 13: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/13.jpg)
![Page 14: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/14.jpg)
Example.
![Page 15: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/15.jpg)
![Page 16: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/16.jpg)
![Page 17: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/17.jpg)
![Page 18: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/18.jpg)
)1)(1()()1()(),()1(
21),(
2
shsshKshsEhhshshhs
s −−−−−Π−
=Π∂∂
)1)(1()()1()(),()1(
21),(
shhhhKhhEhhshshhhs
h −−−−+Π−
=Π∂∂
は微分すると ),( hhsΠ )(),(),,( hEhKhhsΠ がでる.
),( hhsΠ
のとりあつかいについて
![Page 19: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/19.jpg)
)1)(1()()1()(),()1(
21),(
2
shsshKshsEhhshshhs
s −−−−−Π−
=Π∂∂
)1)(1()()1()(),()1(
21),(
shhhhKhhEhhshshhhs
h −−−−+Π−
=Π∂∂
は微分すると ),( hhsΠ )(),(),,( hEhKhhsΠ がでる.
),( hhsΠ
),( hhsΠ
のとりあつかいについて
を消す方法あり!
![Page 20: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/20.jpg)
)1)(1()()1()(),(
)1)(1(2shsss
hKshsEhhss
shss −−
−+−=
Π⋅
−−∂∂
( ))1)(1()1(
)()1()()1(),()1)(1(2
shsshhhKhhEshhs
sshs
h −−−−−−
=
Π⋅
−−∂∂
)1)(1()()1()(),()1(
21),(
2
shsshKshsEhhshshhs
s −−−−−Π−
=Π∂∂
)1)(1()()1()(),()1(
21),(
shhhhKhhEhhshshhhs
h −−−−+Π−
=Π∂∂
は微分すると ),( hhsΠ )(),(),,( hEhKhhsΠ がでる.
),( hhsΠ
),( hhsΠ
),()1)(1(2
hhss
shsΠ⋅
−−
のとりあつかいについて
を消す方法あり!
![Page 21: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/21.jpg)
)1)(1()()1()(),(
)1)(1(2shsss
hKshsEhhss
shss −−
−+−=
−Π⋅
−−∂∂
( ))1)(1()1(
)()1()()1(),()1)(1(2
shsshhhKhhEshhs
sshs
h −−−−−−
=
−Π⋅
−−∂∂
)1)(1()()1()(),()1(
21),(
2
shsshKshsEhhshshhs
s −−−−−−Π−
=−Π∂∂
)1)(1()()1()(),()1(
21),(
shhhhKhhEhhshshhhs
h −−−−+−Π−
=−Π∂∂
は微分すると ),( hhsΠ )(),(),,( hEhKhhsΠ がでる.
),( hhsΠ
),( hhsΠ
),()1)(1(2
hhss
shs−Π⋅
−−
のとりあつかいについて
を消す方法あり!
K.Kosugi, Y. Morita and Y. Yotsutani, DCDS 19(2007),
![Page 22: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/22.jpg)
![Page 23: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/23.jpg)
完全楕円積分 E/K の近似式について
![Page 24: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/24.jpg)
0 1
![Page 25: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/25.jpg)
0 10
0 1
![Page 26: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/26.jpg)
0 10
0 1
![Page 27: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/27.jpg)
)()(
hKhE
の上下からの評価式 )()(
hKhE
![Page 28: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/28.jpg)
・・・
の上下からの評価式 )()(
hKhE
1g
0g
2g
)()(
hKhE・・
1g
0g
2g
)()(
hKhE・・
1
![Page 29: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/29.jpg)
・・・
の上下からの評価式 )()(
hKhE
村井ー松本ー四ッ谷(2010) ガウス(1816)
1g
0g
2g
)()(
hKhE・・
1g
0g
2g
)()(
hKhE・・
1
![Page 30: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/30.jpg)
証明
とおく.
命題 [この場合は直接証明が簡単にできる]
)(hf
![Page 31: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/31.jpg)
![Page 32: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/32.jpg)
![Page 33: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/33.jpg)
![Page 34: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/34.jpg)
![Page 35: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/35.jpg)
![Page 36: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/36.jpg)
![Page 37: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/37.jpg)
![Page 38: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/38.jpg)
![Page 39: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/39.jpg)
![Page 40: 完全楕円積分と楕円関数 - 福岡工業大学h-takeda/conf/files/2015/...sn (z, k) cn (z, k) cn(z,k) →sec h (z,k) as k →1 sn is the Jacobi’s Elliptic function, that](https://reader034.vdocuments.net/reader034/viewer/2022051814/6034e31b7cb1f877f7688c56/html5/thumbnails/40.jpg)
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