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# 16. Elliptic Functions

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## 16.1 Introduction to Elliptic Functions and Integrals

Maximaは Jacobiの楕円関数と不完全楕円積分のサポートを含みます。 これは、数値評価はもちろんこれらの関数のシンボル操作を含みます。 これらの関数の定義とプロパティの多くは Abramowitz and Stegun, 16-17章にあります。 可能な限り、そこで与えられた定義と関係を使います。

 (%i1) jacobi_sn (u, m); (%o1) jacobi_sn(u, m) (%i2) jacobi_sn (u, 1); (%o2) tanh(u) (%i3) jacobi_sn (u, 0); (%o3) sin(u) (%i4) diff (jacobi_sn (u, m), u); (%o4) jacobi_cn(u, m) jacobi_dn(u, m) (%i5) diff (jacobi_sn (u, m), m); (%o5) jacobi_cn(u, m) jacobi_dn(u, m) elliptic_e(asin(jacobi_sn(u, m)), m) (u - ------------------------------------)/(2 m) 1 - m 2 jacobi_cn (u, m) jacobi_sn(u, m) + -------------------------------- 2 (1 - m) 

 (%i1) elliptic_f (phi, m); (%o1) elliptic_f(phi, m) (%i2) elliptic_f (phi, 0); (%o2) phi (%i3) elliptic_f (phi, 1); phi %pi (%o3) log(tan(--- + ---)) 2 4 (%i4) elliptic_e (phi, 1); (%o4) sin(phi) (%i5) elliptic_e (phi, 0); (%o5) phi (%i6) elliptic_kc (1/2); 1 (%o6) elliptic_kc(-) 2 (%i7) makegamma (%); 2 1 gamma (-) 4 (%o7) ----------- 4 sqrt(%pi) (%i8) diff (elliptic_f (phi, m), phi); 1 (%o8) --------------------- 2 sqrt(1 - m sin (phi)) (%i9) diff (elliptic_f (phi, m), m); elliptic_e(phi, m) - (1 - m) elliptic_f(phi, m) (%o9) (----------------------------------------------- m cos(phi) sin(phi) - ---------------------)/(2 (1 - m)) 2 sqrt(1 - m sin (phi)) 

Categories:  Elliptic functions

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## 16.2 Functions and Variables for Elliptic Functions

Jacobiの楕円関数 sn(u,m)

Categories:  Elliptic functions

Jacobiの楕円関数 cn(u,m).

Categories:  Elliptic functions

Jacobiの楕円関数 dn(u,m).

Categories:  Elliptic functions

Jacobiの楕円関数 ns(u,m) = 1/sn(u,m).

Categories:  Elliptic functions

Jacobiの楕円関数 sc(u,m) = sn(u,m)/cn(u,m).

Categories:  Elliptic functions

Jacobiの楕円関数 sd(u,m) = sn(u,m)/dn(u,m).

Categories:  Elliptic functions

Jacobiの楕円関数 nc(u,m) = 1/cn(u,m).

Categories:  Elliptic functions

Jacobiの楕円関数 cs(u,m) = cn(u,m)/sn(u,m).

Categories:  Elliptic functions

Jacobiの楕円関数 cd(u,m) = cn(u,m)/dn(u,m).

Categories:  Elliptic functions

Jacobiの楕円関数 nd(u,m) = 1/dn(u,m).

Categories:  Elliptic functions

Jacobiの楕円関数 ds(u,m) = dn(u,m)/sn(u,m).

Categories:  Elliptic functions

Jacobiの楕円関数 dc(u,m) = dn(u,m)/cn(u,m).

Categories:  Elliptic functions

Jacobiの楕円関数の逆関数 sn(u,m).

Categories:  Elliptic functions

Jacobiの楕円関数の逆関数 cn(u,m).

Categories:  Elliptic functions

Jacobiの楕円関数の逆関数 dn(u,m).

Categories:  Elliptic functions

Jacobiの楕円関数の逆関数 ns(u,m).

Categories:  Elliptic functions

Jacobiの楕円関数の逆関数 sc(u,m).

Categories:  Elliptic functions

Jacobiの楕円関数の逆関数 sd(u,m).

Categories:  Elliptic functions

Jacobiの楕円関数の逆関数 nc(u,m).

Categories:  Elliptic functions

Jacobiの楕円関数の逆関数 cs(u,m).

Categories:  Elliptic functions

Jacobiの楕円関数の逆関数 cd(u,m).

Categories:  Elliptic functions

Jacobiの楕円関数の逆関数 nd(u,m).

Categories:  Elliptic functions

Jacobiの楕円関数の逆関数 ds(u,m).

Categories:  Elliptic functions

Jacobiの楕円関数の逆関数 dc(u,m).

Categories:  Elliptic functions

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## 16.3 Functions and Variables for Elliptic Integrals

integrate(1/sqrt(1 - m*sin(x)^2), x, 0, phi)

elliptic_eelliptic_kcも参照してください。

Categories:  Elliptic integrals

elliptic_e(phi, m) = integrate(sqrt(1 - m*sin(x)^2), x, 0, phi)

elliptic_felliptic_ecも参照してください。

Categories:  Elliptic integrals

integrate(dn(v,m)^2,v,0,u) = integrate(sqrt(1-m*t^2)/sqrt(1-t^2), t, 0, tau)

ここで tau = sn(u,m).

これは

elliptic_eu(u, m) = elliptic_e(asin(sn(u,m)),m) によって elliptic_eと関連付けられます。

elliptic_eも参照してください。

integrate(1/(1-n*sin(x)^2)/sqrt(1 - m*sin(x)^2), x, 0, phi)

Maximaが知っている phiに関する唯一の導関数

Categories:  Elliptic integrals

integrate(1/sqrt(1 - m*sin(x)^2), x, 0, %pi/2)

mのある値に関して 積分の値は Gamma関数で表されることが知られています。 それらを評価するには makegammaを使ってください。

Categories:  Elliptic integrals

integrate(sqrt(1 - m*sin(x)^2), x, 0, %pi/2)

mのある値に関して 積分の値は Gamma関数で表されることが知られています。 それらを評価するには makegammaを使ってください。

Categories:  Elliptic integrals

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