Definicje funkcji Jakobiego
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Funkcje eliptyczne Jakobiego
sn
(
x
,
k
2
)
,
{\displaystyle \operatorname {sn} (x,k^{2}),}
cn
(
x
,
k
2
)
{\displaystyle \operatorname {cn} (x,k^{2})}
dn
(
x
,
k
2
)
{\displaystyle \operatorname {dn} (x,k^{2})}
sn
(
F
(
x
,
k
2
)
,
k
2
)
=
sin
x
;
sn
(
0
,
k
2
)
=
0
{\displaystyle \operatorname {sn} (F(x,k^{2}),k^{2})=\sin x;\quad \operatorname {sn} (0,k^{2})=0}
cn
(
F
(
x
,
k
2
)
,
k
2
)
=
cos
x
;
cn
(
0
,
k
2
)
=
1
{\displaystyle \operatorname {cn} (F(x,k^{2}),k^{2})=\cos x;\quad \operatorname {cn} (0,k^{2})=1}
dn
(
F
(
x
,
k
2
)
,
k
2
)
=
1
−
k
2
sin
2
x
;
dn
(
0
,
k
2
)
=
1
{\displaystyle \operatorname {dn} (F(x,k^{2}),k^{2})={\sqrt {1-k^{2}\sin ^{2}x}};\quad \operatorname {dn} (0,k^{2})=1}
gdzie
F
{\displaystyle F}
niezupełna całka eliptyczna pierwszego rodzaju .
Tw. Funkcje eliptyczne Jakobiego są funkcjami analitycznymi .
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Definiuje się też inne funkcje utworzone z ilorazów funkcji Jakobiego (w analogii do funkcji trygonometrycznych tg x, ctg x, itd.; np. tg x = sin x/ coś x):
ns
(
u
)
=
1
sn
(
u
)
nc
(
u
)
=
1
cn
(
u
)
nd
(
u
)
=
1
dn
(
u
)
sc
(
u
)
=
sn
(
u
)
cn
(
u
)
sd
(
u
)
=
sn
(
u
)
dn
(
u
)
dc
(
u
)
=
dn
(
u
)
cn
(
u
)
ds
(
u
)
=
dn
(
u
)
sn
(
u
)
cs
(
u
)
=
cn
(
u
)
sn
(
u
)
cd
(
u
)
=
cn
(
u
)
dn
(
u
)
{\displaystyle {\begin{aligned}\operatorname {ns} (u)&={\frac {1}{\operatorname {sn} (u)}}\\[8pt]\operatorname {nc} (u)&={\frac {1}{\operatorname {cn} (u)}}\\[8pt]\operatorname {nd} (u)&={\frac {1}{\operatorname {dn} (u)}}\\[8pt]\operatorname {sc} (u)&={\frac {\operatorname {sn} (u)}{\operatorname {cn} (u)}}\\[8pt]\operatorname {sd} (u)&={\frac {\operatorname {sn} (u)}{\operatorname {dn} (u)}}\\[8pt]\operatorname {dc} (u)&={\frac {\operatorname {dn} (u)}{\operatorname {cn} (u)}}\\[8pt]\operatorname {ds} (u)&={\frac {\operatorname {dn} (u)}{\operatorname {sn} (u)}}\\[8pt]\operatorname {cs} (u)&={\frac {\operatorname {cn} (u)}{\operatorname {sn} (u)}}\\[8pt]\operatorname {cd} (u)&={\frac {\operatorname {cn} (u)}{\operatorname {dn} (u)}}\end{aligned}}}
Dla
K
′
=
K
(
1
−
k
2
)
{\displaystyle K'=K(1-k^{2})}
K
=
K
(
1
−
k
2
)
{\displaystyle K=K(1-k^{2})}
K
{\displaystyle K}
zupełna całka eliptyczna pierwszego rodzaju ) można zapisać okresy funkcji:
sn
(
x
,
k
2
)
{\displaystyle \operatorname {sn} (x,k^{2})}
4
K
{\displaystyle 4K}
2
i
K
′
{\displaystyle 2iK'}
cn
(
x
,
k
2
)
{\displaystyle \operatorname {cn} (x,k^{2})}
4
K
{\displaystyle 4K}
2
K
+
2
i
K
′
{\displaystyle 2K+2iK'}
dn
(
x
,
k
2
)
{\displaystyle \operatorname {dn} (x,k^{2})}
2
K
{\displaystyle 2K}
4
i
K
′
{\displaystyle 4iK'}
Funkcje Jacobiego przyjmują wartości rzeczywiste dla
0
<
k
2
<
1
,
{\displaystyle 0<k^{2}<1,}
k
2
=
0
{\displaystyle k^{2}=0}
k
2
=
1
{\displaystyle k^{2}=1}
sn
(
x
,
0
)
=
sin
x
{\displaystyle \operatorname {sn} (x,0)=\sin x}
cn
(
x
,
0
)
=
cos
x
{\displaystyle \operatorname {cn} (x,0)=\cos x}
dn
(
x
,
0
)
=
1
{\displaystyle \operatorname {dn} (x,0)=1}
sn
(
x
,
1
)
=
tgh
x
{\displaystyle \operatorname {sn} (x,1)=\operatorname {tgh} x}
cn
(
x
,
1
)
=
sech
x
{\displaystyle \operatorname {cn} (x,1)=\operatorname {sech} x}
dn
(
x
,
1
)
=
sech
x
{\displaystyle \operatorname {dn} (x,1)=\operatorname {sech} x}
Funkcje te spełniają też następujące zależności:
s
2
+
c
2
=
1
{\displaystyle s^{2}+c^{2}=1}
jedynka trygonometryczna )
k
2
s
2
+
d
2
=
1
{\displaystyle k^{2}s^{2}+d^{2}=1}
gdzie
s
=
sn
(
x
,
k
2
)
,
{\displaystyle s=\operatorname {sn} (x,k^{2}),}
c
=
cn
(
x
,
k
2
)
{\displaystyle c=\operatorname {cn} (x,k^{2})}
d
=
dn
(
x
,
k
2
)
.
{\displaystyle d=\operatorname {dn} (x,k^{2}).}
Ich pochodne dane są przez:
∂
∂
x
sn
(
x
,
k
2
)
=
c
d
{\displaystyle {\frac {\partial }{\partial x}}\operatorname {sn} (x,k^{2})=cd}
∂
∂
x
cn
(
x
,
k
2
)
=
−
s
d
{\displaystyle {\frac {\partial }{\partial x}}\operatorname {cn} (x,k^{2})=-sd}
∂
∂
x
dn
(
x
,
k
2
)
=
−
k
2
s
c
{\displaystyle {\frac {\partial }{\partial x}}\operatorname {dn} (x,k^{2})=-k^{2}sc}
XIII. Elliptic functions and integrals. W: Harry Bateman: Higher transcendental functions . T. II. 1953, s. 294–383. 
![{\displaystyle {\begin{aligned}\operatorname {ns} (u)&={\frac {1}{\operatorname {sn} (u)}}\\[8pt]\operatorname {nc} (u)&={\frac {1}{\operatorname {cn} (u)}}\\[8pt]\operatorname {nd} (u)&={\frac {1}{\operatorname {dn} (u)}}\\[8pt]\operatorname {sc} (u)&={\frac {\operatorname {sn} (u)}{\operatorname {cn} (u)}}\\[8pt]\operatorname {sd} (u)&={\frac {\operatorname {sn} (u)}{\operatorname {dn} (u)}}\\[8pt]\operatorname {dc} (u)&={\frac {\operatorname {dn} (u)}{\operatorname {cn} (u)}}\\[8pt]\operatorname {ds} (u)&={\frac {\operatorname {dn} (u)}{\operatorname {sn} (u)}}\\[8pt]\operatorname {cs} (u)&={\frac {\operatorname {cn} (u)}{\operatorname {sn} (u)}}\\[8pt]\operatorname {cd} (u)&={\frac {\operatorname {cn} (u)}{\operatorname {dn} (u)}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e359bafbfb9a822695ed9dac39e3c61bbb38724)
