자코비 타원함수

개요

\(\text{sn}(z|-1)=z-\frac{z^5}{10}+\frac{z^9}{120}-\frac{11 z^{13}}{15600}+\frac{211 z^{17}}{3536000}+O\left(z^{21}\right)\)

 

 

덧셈공식

\(\begin{align}\operatorname{cn}(x+y) & ={\operatorname{cn}(x)\;\operatorname{cn}(y)- \operatorname{sn}(x)\;\operatorname{sn}(y)\;\operatorname{dn}(x)\;\operatorname{dn}(y)\over {1 - k^2 \;\operatorname{sn}^2 (x) \;\operatorname{sn}^2 (y)}}, \\[8pt]\operatorname{sn}(x+y) & ={\operatorname{sn}(x)\;\operatorname{cn}(y)\;\operatorname{dn}(y) +\operatorname{sn}(y)\;\operatorname{cn}(x)\;\operatorname{dn}(x)\over {1 - k^2 \;\operatorname{sn}^2 (x)\; \operatorname{sn}^2 (y)}}, \\[8pt]\operatorname{dn}(x+y) & ={\operatorname{dn}(x)\;\operatorname{dn}(y)- k^2 \;\operatorname{sn}(x)\;\operatorname{sn}(y)\;\operatorname{cn}(x)\;\operatorname{cn}(y)\over {1 - k^2 \;\operatorname{sn}^2 (x)\; \operatorname{sn}^2 (y)}}.\end{align}\)

 

 

 

메모

 

• Math Overflow http://mathoverflow.net/search?q=

 

 

관련된 항목들

• 렘니스케이트(lemniscate) 곡선의 길이와 타원적분

 

 

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxeVNacEtlVGlYeU0/edit

 

사전 형태의 자료

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Jacobi_elliptic_functions
• The Online Encyclopaedia of Mathematics
• NIST Digital Library of Mathematical Functions
• The World of Mathematical Equations

 

관련논문

• Kiselev, Oleg. “Uniform Asymptotic Behaviour of Jacobi-\(\operatorname{sn}\) near a Singular Point. The Lost Formula from Handbooks for Elliptic Functions.” arXiv:1510.06602 [nlin], October 22, 2015. http://arxiv.org/abs/1510.06602.