[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/jp\/wiki\/archives\/9876#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/jp\/wiki\/archives\/9876","headline":"\u30e9\u30f3\u30c7\u30f3\u5909\u63db – Wikipedia","name":"\u30e9\u30f3\u30c7\u30f3\u5909\u63db – Wikipedia","description":"\u30e9\u30f3\u30c7\u30f3\u5909\u63db (Landen’s transformation) \u306f\u3001\u6570\u5b66\u306b\u304a\u3044\u3066\u6955\u5186\u7a4d\u5206\u3084\u6955\u5186\u95a2\u6570\u306e\u6bcd\u6570\u3092\u5897\u6e1b\u3055\u305b\u308b\u6052\u7b49\u5f0f\u3002\u6955\u5186\u95a2\u6570\u306e\u6570\u5024\u8a08\u7b97\u306b\u6709\u7528\u3067\u3042\u308b\u3002 Table of Contents \u6955\u5186\u7a4d\u5206\u306e\u30e9\u30f3\u30c7\u30f3\u5909\u63db\u3068\u30ac\u30a6\u30b9\u5909\u63db[\u7de8\u96c6]\u30e9\u30f3\u30c7\u30f3\u5909\u63db\u306e\u5c0e\u51fa[\u7de8\u96c6]\u30ac\u30a6\u30b9\u5909\u63db\u306e\u5c0e\u51fa[\u7de8\u96c6]\u6955\u5186\u95a2\u6570\u306e\u30e9\u30f3\u30c7\u30f3\u5909\u63db[\u7de8\u96c6]\u5c0e\u51fa[\u7de8\u96c6]\u865a\u6570\u5909\u63db[\u7de8\u96c6] \u6955\u5186\u7a4d\u5206\u306e\u30e9\u30f3\u30c7\u30f3\u5909\u63db\u3068\u30ac\u30a6\u30b9\u5909\u63db[\u7de8\u96c6] \u7b2c\u4e00\u7a2e\u6955\u5186\u7a4d\u5206 F(sin\u2061\u03b1,k)=\u222bt=0sin\u2061\u03b1dt1\u2212t21\u2212k2t2=\u222b\u03d5=0\u03b1d\u03d51\u2212k2sin2\u2061\u03d5{displaystyle Fleft(sin alpha ,kright)=int _{t=0}^{sin alpha }{frac {dt}{{sqrt {1-t^{2}}}{sqrt {1-k^{2}t^{2}}}}}=int","datePublished":"2020-10-30","dateModified":"2020-10-30","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/jp\/wiki\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/jp\/wiki\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e5887ea658acf46dbb74c4cf592d3a0a0af1c4f2","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e5887ea658acf46dbb74c4cf592d3a0a0af1c4f2","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/jp\/wiki\/archives\/9876","about":["Wiki"],"wordCount":24238,"articleBody":"\u30e9\u30f3\u30c7\u30f3\u5909\u63db (Landen’s transformation) \u306f\u3001\u6570\u5b66\u306b\u304a\u3044\u3066\u6955\u5186\u7a4d\u5206\u3084\u6955\u5186\u95a2\u6570\u306e\u6bcd\u6570\u3092\u5897\u6e1b\u3055\u305b\u308b\u6052\u7b49\u5f0f\u3002\u6955\u5186\u95a2\u6570\u306e\u6570\u5024\u8a08\u7b97\u306b\u6709\u7528\u3067\u3042\u308b\u3002  Table of Contents\u6955\u5186\u7a4d\u5206\u306e\u30e9\u30f3\u30c7\u30f3\u5909\u63db\u3068\u30ac\u30a6\u30b9\u5909\u63db[\u7de8\u96c6]\u30e9\u30f3\u30c7\u30f3\u5909\u63db\u306e\u5c0e\u51fa[\u7de8\u96c6]\u30ac\u30a6\u30b9\u5909\u63db\u306e\u5c0e\u51fa[\u7de8\u96c6]\u6955\u5186\u95a2\u6570\u306e\u30e9\u30f3\u30c7\u30f3\u5909\u63db[\u7de8\u96c6]\u5c0e\u51fa[\u7de8\u96c6]\u865a\u6570\u5909\u63db[\u7de8\u96c6]\u6955\u5186\u7a4d\u5206\u306e\u30e9\u30f3\u30c7\u30f3\u5909\u63db\u3068\u30ac\u30a6\u30b9\u5909\u63db[\u7de8\u96c6]\u7b2c\u4e00\u7a2e\u6955\u5186\u7a4d\u5206  F(sin\u2061\u03b1,k)=\u222bt=0sin\u2061\u03b1dt1\u2212t21\u2212k2t2=\u222b\u03d5=0\u03b1d\u03d51\u2212k2sin2\u2061\u03d5{displaystyle Fleft(sin alpha ,kright)=int _{t=0}^{sin alpha }{frac {dt}{{sqrt {1-t^{2}}}{sqrt {1-k^{2}t^{2}}}}}=int _{phi =0}^{alpha }{frac {dphi }{sqrt {1-k^{2}sin ^{2}phi }}}}\u306b\u3064\u304d\u3001\u6b21\u306e\u6052\u7b49\u5f0f\u3092\u30e9\u30f3\u30c7\u30f3\u5909\u63db\u3068\u3044\u3046\u3002F(sin\u2061\u03b1,k)=21+kF(12(1+k)2sin2\u2061\u03b1+(1\u2212k2sin2\u2061\u03d5\u22121\u2212sin2\u2061\u03d5)2,2k1+k){displaystyle Fleft(sin alpha ,kright)={frac {2}{1+k}}Fleft({frac {1}{2}}{sqrt {left(1+kright)^{2}sin ^{2}alpha +left({sqrt {1-k^{2}sin ^{2}phi }}-{sqrt {1-sin ^{2}phi }}right)^{2}}},{frac {2{sqrt {k}}}{1+k}}right)}  \u540c\u3058\u304f\u3001\u6b21\u306e\u6052\u7b49\u5f0f\u3092\u30ac\u30a6\u30b9\u5909\u63db\u3068\u3044\u3046\u3002F(sin\u2061\u03b1,k)=11+kF((1+k)sin\u2061\u03b11+ksin2\u2061\u03b1,2k1+k){displaystyle Fleft(sin alpha ,kright)={frac {1}{1+k}}Fleft({frac {(1+k)sin alpha }{1+ksin ^{2}alpha }},{frac {2{sqrt {k}}}{1+k}}right)}\u30e9\u30f3\u30c7\u30f3\u5909\u63db\u306e\u5c0e\u51fa[\u7de8\u96c6]\u30e9\u30f3\u30c7\u30f3\u5909\u63db\u306fsin\u2061\u03d5=21+ksin\u2061\u03b8cos\u2061\u03b81\u22124k(1+k)2sin2\u2061\u03b8{displaystyle sin phi ={frac {{frac {2}{1+k}}sin theta cos theta }{sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}}}cos\u2061\u03d5d\u03d5=21+k(cos2\u2061\u03b8\u2212sin2\u2061\u03b8)1\u22124k(1+k)2sin2\u2061\u03b8d\u03b8+21+k(4k(1+k)2sin2\u2061\u03b8cos2\u2061\u03b8)(1\u22124k(1+k)2sin2\u2061\u03b8)3d\u03b8=21+k(1\u221221+ksin2\u2061\u03b8)(1\u22122k1+ksin2\u2061\u03b8)(1\u22124k(1+k)2sin2\u2061\u03b8)3d\u03b8{displaystyle {begin{aligned}cos phi {dphi }&={frac {{frac {2}{1+k}}left(cos ^{2}theta -sin ^{2}theta right)}{sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}}{dtheta }+{frac {{frac {2}{1+k}}left({frac {4k}{(1+k)^{2}}}sin ^{2}theta cos ^{2}theta right)}{left({sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}right)^{3}}}{dtheta }&={frac {{frac {2}{1+k}}left(1-{frac {2}{1+k}}sin ^{2}theta right)left(1-{frac {2k}{1+k}}sin ^{2}theta right)}{left({sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}right)^{3}}}{dtheta }end{aligned}}}\u306e\u7f6e\u63db\u306b\u3088\u308a\u5c0e\u304b\u308c\u308b\u3002F(sin\u2061\u03b1,k)=\u222b\u03d5=0\u03b1d\u03d51\u2212k2sin2\u2061\u03d5=\u222b\u03d5=0\u03b1cos\u2061\u03d5d\u03d51\u2212sin2\u2061\u03d51\u2212k2sin2\u2061\u03d5=\u222b\u03b8=0\u03b221+k(1\u221221+ksin2\u2061\u03b8)(1\u22122k1+ksin2\u2061\u03b8)(1\u22124k(1+k)2sin2\u2061\u03b8)31\u22124(1+k)2sin2\u2061\u03b8cos2\u2061\u03b81\u22124k(1+k)2sin2\u2061\u03b81\u2212k24(1+k)2sin2\u2061\u03b8cos2\u2061\u03b81\u22124k(1+k)2sin2\u2061\u03b8d\u03b8=\u222b\u03b8=0\u03b221+k(1\u221221+ksin2\u2061\u03b8)(1\u22122k1+ksin2\u2061\u03b8)(1\u22124k(1+k)2sin2\u2061\u03b8)31\u221221+ksin2\u2061\u03b81\u22124k(1+k)2sin2\u2061\u03b81\u22122k1+ksin2\u2061\u03b81\u22124k(1+k)2sin2\u2061\u03b8d\u03b8=21+k\u222b\u03b8=0\u03b2d\u03b81\u22124k(1+k)2sin2\u2061\u03b8=21+kF(sin\u2061\u03b2,2k1+k){displaystyle {begin{aligned}Fleft(sin alpha ,kright)&=int _{phi =0}^{alpha }{frac {dphi }{sqrt {1-k^{2}sin ^{2}phi }}}&=int _{phi =0}^{alpha }{frac {cos phi {dphi }}{{sqrt {1-sin ^{2}phi }}{sqrt {1-k^{2}sin ^{2}phi }}}}&=int _{theta =0}^{beta }{frac {frac {{frac {2}{1+k}}left(1-{frac {2}{1+k}}sin ^{2}theta right)left(1-{frac {2k}{1+k}}sin ^{2}theta right)}{left({sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}right)^{3}}}{{sqrt {1-{frac {{frac {4}{(1+k)^{2}}}sin ^{2}theta cos ^{2}theta }{1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}}}{sqrt {1-k^{2}{frac {{frac {4}{(1+k)^{2}}}sin ^{2}theta cos ^{2}theta }{1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}}}}}{dtheta }&=int _{theta =0}^{beta }{frac {frac {{frac {2}{1+k}}left(1-{frac {2}{1+k}}sin ^{2}theta right)left(1-{frac {2k}{1+k}}sin ^{2}theta right)}{left({sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}right)^{3}}}{{frac {1-{frac {2}{1+k}}sin ^{2}theta }{sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}};{frac {1-{frac {2k}{1+k}}sin ^{2}theta }{sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}}}}{dtheta }&={frac {2}{1+k}}int _{theta =0}^{beta }{frac {dtheta }{sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}}&={frac {2}{1+k}}Fleft(sin beta ,{frac {2{sqrt {k}}}{1+k}}right)end{aligned}}}sin\u2061\u03b2{displaystyle sin beta }\u3092\u967d\u306b\u3059\u308b\u3068\u3067\u3042\u308b\u3002\u30ac\u30a6\u30b9\u5909\u63db\u306e\u5c0e\u51fa[\u7de8\u96c6]\u30ac\u30a6\u30b9\u5909\u63db\u306fsin\u2061\u03d5=21+ksin\u2061\u03b81+1\u22124k(1+k)2sin2\u2061\u03b8{displaystyle sin phi ={frac {{frac {2}{1+k}}sin theta }{1+{sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}}}}cos\u2061\u03d5d\u03d5=21+kcos\u2061\u03b81+1\u22124k(1+k)2sin2\u2061\u03b8d\u03b8+21+k(4k(1+k)2sin2\u2061\u03b8cos\u2061\u03b8)1\u22124k(1+k)2sin2\u2061\u03b8(1+1\u22124k(1+k)2sin2\u2061\u03b8)2d\u03b8=21+kcos\u2061\u03b81\u22124k(1+k)2sin2\u2061\u03b8(1+1\u22124k(1+k)2sin2\u2061\u03b8)d\u03b8{displaystyle {begin{aligned}cos phi {dphi }&={frac {{frac {2}{1+k}}cos theta }{1+{sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}}}{dtheta }+{frac {{frac {2}{1+k}}left({frac {4k}{(1+k)^{2}}}sin ^{2}theta cos theta right)}{{sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}left(1+{sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}right)^{2}}}{dtheta }&={frac {{frac {2}{1+k}}cos theta }{{sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}left(1+{sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}right)}}{dtheta }end{aligned}}}\u306e\u7f6e\u63db\u306b\u3088\u308a\u5c0e\u304b\u308c\u308b\u3002F(\u03b1,k)=\u222b\u03d5=0\u03b1d\u03d51\u2212k2sin2\u2061\u03d5=\u222b\u03d5=0\u03b1cos\u2061\u03d5d\u03d51\u2212sin2\u2061\u03d51\u2212k2sin2\u2061\u03d5=\u222b\u03b8=0\u03b221+kcos\u2061\u03b81\u22124k(1+k)2sin2\u2061\u03b8(1+1\u22124k(1+k)2sin2\u2061\u03b8)2+21\u22124k(1+k)2sin2\u2061\u03b8\u221241+ksin2\u2061\u03b81+1\u22124k(1+k)2sin2\u2061\u03b82+21\u22124k(1+k)2sin2\u2061\u03b8\u22124k1+ksin2\u2061\u03b81+1\u22124k(1+k)2sin2\u2061\u03b8d\u03b8=\u222b\u03b8=0\u03b221+kcos\u2061\u03b81\u22124k(1+k)2sin2\u2061\u03b8(1+1\u22124k(1+k)2sin2\u2061\u03b8)21\u2212sin2\u2061\u03b82+21\u22124k(1+k)2sin2\u2061\u03b8\u22124k(1+k)2sin2\u2061\u03b8(1+1\u22124k(1+k)2sin2\u2061\u03b8)2d\u03b8=11+k\u222b\u03b8=0\u03b211\u22124k(1+k)2sin2\u2061\u03b8=11+kF(\u03b2,2k1+k){displaystyle {begin{aligned}Fleft(alpha ,kright)&=int _{phi =0}^{alpha }{frac {dphi }{sqrt {1-k^{2}sin ^{2}phi }}}&=int _{phi =0}^{alpha }{frac {cos phi {dphi }}{{sqrt {1-sin ^{2}phi }}{sqrt {1-k^{2}sin ^{2}phi }}}}&=int _{theta =0}^{beta }{frac {frac {{frac {2}{1+k}}cos theta }{{sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}left(1+{sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}right)}}{{frac {sqrt {2+2{sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}-{frac {4}{1+k}}sin ^{2}theta }}{1+{sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}}};{frac {sqrt {2+2{sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}-{frac {4k}{1+k}}sin ^{2}theta }}{1+{sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}}}}}{dtheta }&=int _{theta =0}^{beta }{frac {frac {{frac {2}{1+k}}cos theta }{{sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}left(1+{sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}right)}}{frac {2{sqrt {1-sin ^{2}theta }}{sqrt {2+2{sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}}{left(1+{sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}right)^{2}}}}{dtheta }&={frac {1}{1+k}}int _{theta =0}^{beta }{frac {1}{sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}theta }}}&={frac {1}{1+k}}Fleft(beta ,{frac {2{sqrt {k}}}{1+k}}right)end{aligned}}}sin\u2061\u03b2{displaystyle sin beta }\u3092\u967d\u306b\u3059\u308b\u3068\u3067\u3042\u308b\u3002\u6955\u5186\u95a2\u6570\u306e\u30e9\u30f3\u30c7\u30f3\u5909\u63db[\u7de8\u96c6]\u6b21\u306e\u6052\u7b49\u5f0f\u3092\u6955\u5186\u95a2\u6570\u306e\u4e0a\u6607\u30e9\u30f3\u30c7\u30f3\u5909\u63db\u3068\u3044\u3046\u3002sn\u2061(u,k)=21+ksn\u2061(1+k2u,2k1+k)cn\u2061(1+k2u,2k1+k)dn\u2061(1+k2u,2k1+k){displaystyle operatorname {sn} left(u,kright)={frac {{tfrac {2}{1+k}}operatorname {sn} left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)operatorname {cn} left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)}{operatorname {dn} left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)}}}cn\u2061(u,k)=21+kdn2\u2061(1+k2u,2k1+k)\u22121\u2212k1+k4k(1+k)2dn\u2061(1+k2u,2k1+k){displaystyle operatorname {cn} left(u,kright)={frac {{tfrac {2}{1+k}}operatorname {dn} ^{2}left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)-{tfrac {1-k}{1+k}}}{{tfrac {4k}{(1+k)^{2}}}operatorname {dn} left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)}}}dn\u2061(u,k)=2k1+kdn2\u2061(1+k2u,2k1+k)+1\u2212k1+k4k(1+k)2dn\u2061(1+k2u,2k1+k){displaystyle operatorname {dn} left(u,kright)={frac {{tfrac {2k}{1+k}}operatorname {dn} ^{2}left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)+{tfrac {1-k}{1+k}}}{{tfrac {4k}{(1+k)^{2}}}operatorname {dn} left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)}}}\u6b21\u306e\u6052\u7b49\u5f0f\u3092\u6955\u5186\u95a2\u6570\u306e\u4e0b\u964d\u30e9\u30f3\u30c7\u30f3\u5909\u63db\u3068\u3044\u3046\u3002sn\u2061(u,k)=21+1\u2212k2sn\u2061(1+1\u2212k22u,1\u22121\u2212k21+1\u2212k2)1+1\u22121\u2212k21+1\u2212k2sn2\u2061(1+1\u2212k22u,1\u22121\u2212k21+1\u2212k2){displaystyle operatorname {sn} left(u,kright)={frac {{tfrac {2}{1+{sqrt {1-k^{2}}}}}operatorname {sn} left({tfrac {1+{sqrt {1-k^{2}}}}{2}}u,{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}right)}{1+{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}operatorname {sn} ^{2}left({tfrac {1+{sqrt {1-k^{2}}}}{2}}u,{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}right)}}}cn\u2061(u,k)=cn\u2061(1+1\u2212k22u,1\u22121\u2212k21+1\u2212k2)dn\u2061(1+1\u2212k22u,1\u22121\u2212k21+1\u2212k2)1+1\u22121\u2212k21+1\u2212k2sn2\u2061(1+1\u2212k22u,1\u22121\u2212k21+1\u2212k2){displaystyle operatorname {cn} left(u,kright)={frac {operatorname {cn} left({tfrac {1+{sqrt {1-k^{2}}}}{2}}u,{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}right)operatorname {dn} left({tfrac {1+{sqrt {1-k^{2}}}}{2}}u,{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}right)}{1+{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}operatorname {sn} ^{2}left({tfrac {1+{sqrt {1-k^{2}}}}{2}}u,{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}right)}}}dn\u2061(u,k)=1\u22121\u2212k21+1\u2212k2\u2212(1\u2212dn2\u2061(1+1\u2212k22u,1\u22121\u2212k21+1\u2212k2))1\u22121\u2212k21+1\u2212k2+(1\u2212dn2\u2061(1+1\u2212k22u,1\u22121\u2212k21+1\u2212k2)){displaystyle operatorname {dn} left(u,kright)={frac {{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}-left(1-operatorname {dn} ^{2}left({tfrac {1+{sqrt {1-k^{2}}}}{2}}u,{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}right)right)}{{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}+left(1-operatorname {dn} ^{2}left({tfrac {1+{sqrt {1-k^{2}}}}{2}}u,{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}right)right)}}}\u5f53\u521d\u306e\u6bcd\u6570\u304c0\u03b2cos\u2061\u03b21\u22124k(1+k)2sin2\u2061\u03b2{displaystyle sin alpha ={frac {{frac {2}{1+k}}sin beta cos beta }{sqrt {1-{frac {4k}{(1+k)^{2}}}sin ^{2}beta }}}}\u306e\u3068\u304d\u306bu=F(\u03b1,k)=21+kF(\u03b2,2k1+k){displaystyle u=Fleft(alpha ,kright)={tfrac {2}{1+k}}Fleft(beta ,{tfrac {2{sqrt {k}}}{1+k}}right)}sn\u2061(u,k)=sin\u2061\u03b1{displaystyle operatorname {sn} left(u,kright)=sin alpha }sn\u2061(1+k2u,2k1+k)=sin\u2061\u03b2{displaystyle operatorname {sn} left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)=sin beta }\u3067\u3042\u308b\u304b\u3089sn\u2061(u,k)=21+ksn\u2061(1+k2u,2k1+k)1\u2212sn2\u2061(1+k2u,2k1+k)1\u2212(2k1+k)2sn2\u2061(1+k2u,2k1+k)=21+ksn\u2061(1+k2u,2k1+k)cn\u2061(1+k2u,2k1+k)dn\u2061(1+k2u,2k1+k){displaystyle operatorname {sn} left(u,kright)={frac {{tfrac {2}{1+k}}operatorname {sn} left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right){sqrt {1-operatorname {sn} ^{2}left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)}}}{sqrt {1-left({tfrac {2{sqrt {k}}}{1+k}}right)^{2}operatorname {sn} ^{2}left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)}}}={frac {{tfrac {2}{1+k}}operatorname {sn} left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)operatorname {cn} left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)}{operatorname {dn} left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)}}}cn\u2061(u,k)=1\u2212sn2\u2061(u,k)=1\u221221+ksn2\u2061(1+k2u,2k1+k)dn\u2061(1+k2u,2k1+k)=21+kdn2\u2061(1+k2u,2k1+k)\u22121\u2212k1+k4k(1+k)2dn\u2061(1+k2u,2k1+k){displaystyle operatorname {cn} left(u,kright)={sqrt {1-operatorname {sn} ^{2}left(u,kright)}}={frac {1-{tfrac {2}{1+k}}operatorname {sn} ^{2}left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)}{operatorname {dn} left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)}}={frac {{tfrac {2}{1+k}}operatorname {dn} ^{2}left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)-{tfrac {1-k}{1+k}}}{{tfrac {4k}{(1+k)^{2}}}operatorname {dn} left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)}}}dn\u2061(u,k)=1\u2212k2sn2\u2061(u,k)=1\u22122k1+ksn2\u2061(1+k2u,2k1+k)dn\u2061(1+k2u,2k1+k)=2k1+kdn2\u2061(1+k2u,2k1+k)+1\u2212k1+k4k(1+k)2dn\u2061(1+k2u,2k1+k){displaystyle operatorname {dn} left(u,kright)={sqrt {1-k^{2}operatorname {sn} ^{2}left(u,kright)}}={frac {1-{tfrac {2k}{1+k}}operatorname {sn} ^{2}left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)}{operatorname {dn} left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)}}={frac {{tfrac {2k}{1+k}}operatorname {dn} ^{2}left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)+{tfrac {1-k}{1+k}}}{{tfrac {4k}{(1+k)^{2}}}operatorname {dn} left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)}}}\u3067\u3042\u308b\u3002\u6955\u5186\u7a4d\u5206\u306e\u30ac\u30a6\u30b9\u5909\u63db\u306b\u3088\u308asin\u2061\u03b2=(1+k)sin\u2061\u03b11+ksin2\u2061\u03b1{displaystyle sin beta ={frac {(1+k)sin alpha }{1+ksin ^{2}alpha }}}\u306e\u3068\u304d\u306bu=F(\u03b1,k)=11+kF(\u03b2,2k1+k){displaystyle u=Fleft(alpha ,kright)={tfrac {1}{1+k}}Fleft(beta ,{tfrac {2{sqrt {k}}}{1+k}}right)}sn\u2061(u,k)=sin\u2061\u03b1{displaystyle operatorname {sn} left(u,kright)=sin alpha }sn\u2061((1+k)u,2k1+k)=sin\u2061\u03b2{displaystyle operatorname {sn} left((1+k)u,{tfrac {2{sqrt {k}}}{1+k}}right)=sin beta }\u3067\u3042\u308b\u304b\u3089sn\u2061((1+k)u,2k1+k)=(1+k)sn\u2061\u03b11+ksn2\u2061\u03b1{displaystyle operatorname {sn} left((1+k)u,{tfrac {2{sqrt {k}}}{1+k}}right)={frac {(1+k)operatorname {sn} alpha }{1+koperatorname {sn} ^{2}alpha }}}\u3067\u3042\u308b\u304c\u3001u{displaystyle u}\u3092u1+k{displaystyle {tfrac {u}{1+k}}}\u306b\u6539\u3081\u3001k{displaystyle k}\u30921\u22121\u2212k21+1\u2212k2{displaystyle {tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}}\u306b\u6539\u3081\u308c\u3070sn\u2061(u,k)=21+1\u2212k2sn\u2061(1+1\u2212k22u,1\u22121\u2212k21+1\u2212k2)1+1\u22121\u2212k21+1\u2212k2sn2\u2061(1+1\u2212k22u,1\u22121\u2212k21+1\u2212k2){displaystyle operatorname {sn} left(u,kright)={frac {{tfrac {2}{1+{sqrt {1-k^{2}}}}}operatorname {sn} left({tfrac {1+{sqrt {1-k^{2}}}}{2}}u,{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}right)}{1+{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}operatorname {sn} ^{2}left({tfrac {1+{sqrt {1-k^{2}}}}{2}}u,{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}right)}}}cn\u2061(u,k)=1\u2212sn2\u2061(u,k)=cn\u2061(1+1\u2212k22u,1\u22121\u2212k21+1\u2212k2)dn\u2061(1+1\u2212k22u,1\u22121\u2212k21+1\u2212k2)1+1\u22121\u2212k21+1\u2212k2sn2\u2061(1+1\u2212k22u,1\u22121\u2212k21+1\u2212k2){displaystyle {begin{aligned}operatorname {cn} left(u,kright)&={sqrt {1-operatorname {sn} ^{2}left(u,kright)}}&={frac {operatorname {cn} left({tfrac {1+{sqrt {1-k^{2}}}}{2}}u,{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}right)operatorname {dn} left({tfrac {1+{sqrt {1-k^{2}}}}{2}}u,{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}right)}{1+{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}operatorname {sn} ^{2}left({tfrac {1+{sqrt {1-k^{2}}}}{2}}u,{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}right)}}end{aligned}}}dn\u2061(u,k)=1\u2212k2sn2\u2061(u,k)=1\u22121\u22121\u2212k21+1\u2212k2sn2\u2061(1+1\u2212k22u,1\u22121\u2212k21+1\u2212k2)1+1\u22121\u2212k21+1\u2212k2sn2\u2061(1+1\u2212k22u,1\u22121\u2212k21+1\u2212k2)=dn2\u2061(1+1\u2212k22u,1\u22121\u2212k21+1\u2212k2)\u221221\u2212k21+1\u2212k221+1\u2212k2\u2212dn2\u2061(1+1\u2212k22u,1\u22121\u2212k21+1\u2212k2){displaystyle {begin{aligned}operatorname {dn} left(u,kright)&={sqrt {1-k^{2}operatorname {sn} ^{2}left(u,kright)}}&={frac {1-{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}operatorname {sn} ^{2}left({tfrac {1+{sqrt {1-k^{2}}}}{2}}u,{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}right)}{1+{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}operatorname {sn} ^{2}left({tfrac {1+{sqrt {1-k^{2}}}}{2}}u,{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}right)}}&={frac {operatorname {dn} ^{2}left({tfrac {1+{sqrt {1-k^{2}}}}{2}}u,{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}right)-{tfrac {2{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}}{{tfrac {2}{1+{sqrt {1-k^{2}}}}}-operatorname {dn} ^{2}left({tfrac {1+{sqrt {1-k^{2}}}}{2}}u,{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}right)}}end{aligned}}}\u3068\u306a\u308b\u3002\u865a\u6570\u5909\u63db[\u7de8\u96c6]\u4e0a\u6607\u30e9\u30f3\u30c7\u30f3\u5909\u63db\u3068\u4e0b\u964d\u30e9\u30f3\u30c7\u30f3\u5909\u63db\u306f\u865a\u6570\u5909\u63db\u306b\u3088\u308a\u4ea4\u66ff\u3059\u308b\u3002sn\u2061(iu,1\u2212k2)=isc\u2061(u,k)=isn\u2061(u,k)cn\u2061(u,k){displaystyle operatorname {sn} left(iu,{sqrt {1-k^{2}}}right)=ioperatorname {sc} left(u,kright)={frac {ioperatorname {sn} left(u,kright)}{operatorname {cn} left(u,kright)}}}\u4e0a\u6607\u30e9\u30f3\u30c7\u30f3\u5909\u63db\u306b\u3088\u308aisn\u2061(u,k)cn\u2061(u,k)=2i1+ksn\u2061(1+k2u,2k1+k)cn\u2061(1+k2u,2k1+k)dn\u2061(1+k2u,2k1+k)21+kdn2\u2061(1+k2u,2k1+k)\u22121\u2212k1+k4k(1+k)2dn\u2061(1+k2u,2k1+k)=4ki(1+k)2sn\u2061(1+k2u,2k1+k)cn\u2061(1+k2u,2k1+k)dn2\u2061(1+k2u,2k1+k)\u22121\u2212k1+k{displaystyle {begin{aligned}{frac {ioperatorname {sn} left(u,kright)}{operatorname {cn} left(u,kright)}}&={frac {frac {{tfrac {2i}{1+k}}operatorname {sn} left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)operatorname {cn} left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)}{operatorname {dn} left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)}}{frac {{tfrac {2}{1+k}}operatorname {dn} ^{2}left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)-{tfrac {1-k}{1+k}}}{{tfrac {4k}{(1+k)^{2}}}operatorname {dn} left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)}}}&={frac {{tfrac {4ki}{(1+k)^{2}}}operatorname {sn} left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)operatorname {cn} left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)}{operatorname {dn} ^{2}left({tfrac {1+k}{2}}u,{tfrac {2{sqrt {k}}}{1+k}}right)-{tfrac {1-k}{1+k}}}}end{aligned}}}\u865a\u6570\u5909\u63db\u306b\u3088\u308asn\u2061(iu,1\u2212k2)=4k(1+k)2sc\u2061(1+k2iu,1\u2212k1+k)nc\u2061(1+k2iu,1\u2212k1+k)dc2\u2061(1+k2iu,1\u2212k1+k)\u22121\u2212k1+k=4k(1+k)2sn\u2061(1+k2iu,1\u2212k1+k)dn2\u2061(1+k2iu,1\u2212k1+k)\u22121\u2212k1+kcn2\u2061(1+k2iu,1\u2212k1+k)=4k(1+k)2sn\u2061(1+k2iu,1\u2212k1+k)2k1+k+2k(1\u2212k)(1+k)2sn2\u2061(1+k2iu,1\u2212k1+k)=21+ksn\u2061(1+k2iu,1\u2212k1+k)1+1\u2212k1+ksn2\u2061(1+k2iu,1\u2212k1+k){displaystyle {begin{aligned}operatorname {sn} left(iu,{sqrt {1-k^{2}}}right)&={frac {{tfrac {4k}{(1+k)^{2}}}operatorname {sc} left({tfrac {1+k}{2}}iu,{tfrac {1-k}{1+k}}right)operatorname {nc} left({tfrac {1+k}{2}}iu,{tfrac {1-k}{1+k}}right)}{operatorname {dc} ^{2}left({tfrac {1+k}{2}}iu,{tfrac {1-k}{1+k}}right)-{tfrac {1-k}{1+k}}}}&={frac {{tfrac {4k}{(1+k)^{2}}}operatorname {sn} left({tfrac {1+k}{2}}iu,{tfrac {1-k}{1+k}}right)}{operatorname {dn} ^{2}left({tfrac {1+k}{2}}iu,{tfrac {1-k}{1+k}}right)-{tfrac {1-k}{1+k}}operatorname {cn} ^{2}left({tfrac {1+k}{2}}iu,{tfrac {1-k}{1+k}}right)}}&={frac {{tfrac {4k}{(1+k)^{2}}}operatorname {sn} left({tfrac {1+k}{2}}iu,{tfrac {1-k}{1+k}}right)}{{tfrac {2k}{1+k}}+{tfrac {2k(1-k)}{(1+k)^{2}}}operatorname {sn} ^{2}left({tfrac {1+k}{2}}iu,{tfrac {1-k}{1+k}}right)}}&={frac {{tfrac {2}{1+k}}operatorname {sn} left({tfrac {1+k}{2}}iu,{tfrac {1-k}{1+k}}right)}{1+{tfrac {1-k}{1+k}}operatorname {sn} ^{2}left({tfrac {1+k}{2}}iu,{tfrac {1-k}{1+k}}right)}}end{aligned}}}iu{displaystyle iu}\u3092u{displaystyle u}\u3068\u66f8\u304d\u30011\u2212k2{displaystyle {sqrt {1-k^{2}}}}\u3092k{displaystyle k}\u3068\u66f8\u3051\u3070sn\u2061(u,k)=21+1\u2212k2sn\u2061(1+1\u2212k22u,1\u22121\u2212k21+1\u2212k2)1+1\u22121\u2212k21+1\u2212k2sn2\u2061(1+1\u2212k22u,1\u22121\u2212k21+1\u2212k2){displaystyle {begin{aligned}operatorname {sn} left(u,kright)&={frac {{tfrac {2}{1+{sqrt {1-k^{2}}}}}operatorname {sn} left({tfrac {1+{sqrt {1-k^{2}}}}{2}}u,{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}right)}{1+{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}operatorname {sn} ^{2}left({tfrac {1+{sqrt {1-k^{2}}}}{2}}u,{tfrac {1-{sqrt {1-k^{2}}}}{1+{sqrt {1-k^{2}}}}}right)}}end{aligned}}}\u3068\u306a\u308b\u304c\u3001\u3053\u308c\u306f\u4e0b\u964d\u30e9\u30f3\u30c7\u30f3\u5909\u63db\u3067\u3042\u308b\u3002"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki\/archives\/9876#breadcrumbitem","name":"\u30e9\u30f3\u30c7\u30f3\u5909\u63db – Wikipedia"}}]}]