{"version":10,"randomSeed":"bcc02434e5f9d9a12579c28d23b1e77d","graph":{"viewport":{"xmin":-108.43066941763601,"ymin":-111.32904434123455,"xmax":163.64794140802792,"ymax":133.82390110676366}},"expressions":{"list":[{"type":"text","id":"22","text":"inspired by \"Winding numbers and domain coloring\" by 3Blue1Brown (2018) https://www.youtube.com/watch?v=b7FxPsqfkOY"},{"type":"folder","id":"2","title":"basic functions of complex numbers","collapsed":true},{"type":"text","id":"3","folderId":"2","text":"multiply"},{"type":"expression","id":"4","folderId":"2","color":"#c74440","latex":"M_{c}\\left(w,z\\right)=\\left(w.x\\cdot z.x-w.y\\cdot z.y,w.x\\cdot z.y+w.y\\cdot z.x\\right)"},{"type":"text","id":"5","folderId":"2","text":"reciprocate"},{"type":"expression","id":"6","folderId":"2","color":"#388c46","latex":"R_{c}\\left(z\\right)=\\frac{\\left(z.x,-z.y\\right)}{z.x^{2}+z.y^{2}}"},{"type":"text","id":"7","folderId":"2","text":"square"},{"type":"expression","id":"8","folderId":"2","color":"#2d70b3","latex":"Q_{c}\\left(z\\right)=M_{c}\\left(z,z\\right)"},{"type":"text","id":"9","folderId":"2","text":"cube"},{"type":"expression","id":"10","folderId":"2","color":"#000000","latex":"C_{c}\\left(z\\right)=M_{c}\\left(Q_{c}\\left(z\\right),z\\right)"},{"type":"text","id":"11","folderId":"2","text":"naturally exponentiate"},{"type":"expression","id":"12","folderId":"2","color":"#c74440","latex":"E_{c}\\left(z\\right)=\\left(e^{z.x}\\cos\\left(z.y\\right),e^{z.x}\\sin\\left(z.y\\right)\\right)"},{"type":"text","id":"13","folderId":"2","text":"naturally logarithmise"},{"type":"expression","id":"14","folderId":"2","color":"#2d70b3","latex":"L_{c}\\left(z\\right)=\\left(\\ln\\left(\\operatorname{distance}\\left(\\left(0,0\\right),z\\right)\\right),\\arctan\\left(z.y,z.x\\right)\\right)"},{"type":"text","id":"15","folderId":"2","text":"power (complex to real)"},{"type":"expression","id":"16","folderId":"2","color":"#2d70b3","latex":"P_{c}\\left(z,x\\right)=\\left(\\operatorname{distance}\\left(\\left(0,0\\right),z\\right)^{x}\\cos\\left(x\\arctan\\left(z.y,z.x\\right)\\right),\\operatorname{distance}\\left(\\left(0,0\\right),z\\right)^{x}\\sin\\left(x\\arctan\\left(z.y,z.x\\right)\\right)\\right)"},{"type":"text","id":"40","text":"function to consider"},{"type":"expression","id":"23","color":"#c74440","latex":"F\\left(z\\right)=2M_{c}\\left(E_{c}\\left(z\\right),M_{c}\\left(\\left(\\frac{1}{2},0\\right),E_{c}\\left(M_{c}\\left(\\left(0,1\\right),z\\right)\\right)+E_{c}\\left(M_{c}\\left(\\left(0,-1\\right),z\\right)\\right)\\right)\\right)+\\left(1,0\\right)"},{"type":"expression","id":"73","color":"#000000","latex":"f\\left(x,y\\right)=F\\left(\\left(x,y\\right)\\right)"},{"type":"text","id":"42","text":"derivative"},{"type":"expression","id":"27","color":"#000000","latex":"g\\left(x,y\\right)=\\frac{d}{dx}f\\left(x,y\\right)"},{"type":"expression","id":"77","color":"#6042a6","latex":"G\\left(z\\right)=g\\left(z.x,z.y\\right)"},{"type":"text","id":"44","text":"logarithmic derivative used in argument principle integral"},{"type":"expression","id":"28","color":"#c74440","latex":"A_{rg}\\left(z\\right)=M_{c}\\left(G\\left(z\\right),R_{c}\\left(F\\left(z\\right)\\right)\\right)"},{"type":"folder","id":"65","title":"list of boxes (by corners) to integrate around"},{"type":"expression","id":"17","folderId":"65","color":"#c74440","latex":"Q=\\left[\\left(1.665534395026359,0\\right),\\left(4.708139185299269,0\\right),\\left(7.854454682042923,0\\right),\\left(10.995844170452136,0\\right),\\left(14.137445605327924,0\\right),\\left(17.27903787944716,0\\right),\\left(20.42063054943537,0\\right),\\left(23.56222320231652,0\\right),\\left(26.703815855936938,0\\right),\\left(29.84540850952541,0\\right),\\left(32.98700116311526,0\\right),\\left(36.12859381670505,0\\right),\\left(39.270186470294846,0\\right),\\left(42.411779123884635,0\\right),\\left(45.55337177747443,0\\right),\\left(48.69496443106422,0\\right),\\left(51.83655708465402,0\\right),\\left(54.97814973824381,0\\right),\\left(58.119742391833604,0\\right),\\left(61.261335045423394,0\\right),\\left(64.40292769901319,0\\right),\\left(67.54452035260299,0\\right),\\left(70.68611300619277,0\\right),\\left(73.82770565978257,0\\right),\\left(76.96929831337236,0\\right),\\left(80.11089096696216,0\\right),\\left(83.25248362055194,0\\right),\\left(86.39407627414174,0\\right),\\left(89.53566892773154,0\\right),\\left(92.67726158132133,0\\right),\\left(95.81885423491111,0\\right),\\left(98.96044688850091,0\\right),\\left(102.10203954209071,0\\right),\\left(105.2436321956805,0\\right),\\left(108.38522484927029,0\\right),\\left(111.52681750286008,0\\right),\\left(114.66841015644988,0\\right),\\left(117.81000281003968,0\\right),\\left(120.95159546362947,0\\right),\\left(124.09318811721926,0\\right),\\left(-1.5505692805699554,1.592768030477778\\right),\\left(-4.712348635510131,4.712429331770864\\right),\\left(-7.853981558623642,7.853981709325339\\right),\\left(-10.995574287423567,10.995574287704976\\right),\\left(-14.137166941153811,14.13716694115432\\right),\\left(-17.278759594743867,17.27875959474385\\right),\\left(-20.42035224833366,20.420352248333643\\right),\\left(-23.561944901923454,23.561944901923436\\right),\\left(-26.703537555513247,26.70353755551323\\right),\\left(-29.84513020910304,29.845130209103022\\right),\\left(-32.98672286269284,32.986722862692815\\right),\\left(-36.128315516282626,36.12831551628261\\right),\\left(-39.26990816987242,39.2699081698724\\right),\\left(-42.41150082346221,42.4115008234622\\right),\\left(-45.55309347705201,45.55309347705199\\right),\\left(-48.6946861306418,48.694686130641784\\right),\\left(-51.836278784231595,51.836278784231574\\right),\\left(-54.977871437821385,54.97787143782137\\right),\\left(-58.11946409141118,58.11946409141116\\right),\\left(-61.26105674500097,61.26105674500096\\right),\\left(-64.40264939859077,64.40264939859075\\right),\\left(-67.54424205218056,67.54424205218054\\right),\\left(-70.68583470577035,70.68583470577033\\right),\\left(-73.82742735936014,73.82742735936013\\right),\\left(-76.96902001294994,76.96902001294993\\right),\\left(-80.11061266653974,80.11061266653971\\right),\\left(-83.25220532012952,83.2522053201295\\right),\\left(-86.39379797371932,86.3937979737193\\right),\\left(-89.53539062730911,89.5353906273091\\right),\\left(-92.67698328089891,92.67698328089888\\right),\\left(-95.81857593448869,95.81857593448868\\right),\\left(-98.96016858807849,98.96016858807847\\right),\\left(-102.10176124166829,102.10176124166827\\right),\\left(-105.24335389525808,105.24335389525805\\right),\\left(-108.38494654884788,108.38494654884785\\right),\\left(-111.52653920243766,111.52653920243765\\right),\\left(-114.66813185602746,114.66813185602744\\right),\\left(-117.80972450961725,117.80972450961724\\right),\\left(-120.95131716320705,120.95131716320702\\right),\\left(-124.09290981679683,124.09290981679682\\right),\\left(-1.5505692805699554,-1.592768030477778\\right),\\left(-4.712348635510131,-4.712429331770864\\right),\\left(-7.853981558623642,-7.853981709325339\\right),\\left(-10.995574287423567,-10.995574287704976\\right),\\left(-14.137166941153811,-14.13716694115432\\right),\\left(-17.278759594743867,-17.27875959474385\\right),\\left(-20.42035224833366,-20.420352248333643\\right),\\left(-23.561944901923454,-23.561944901923436\\right),\\left(-26.703537555513247,-26.70353755551323\\right),\\left(-29.84513020910304,-29.845130209103022\\right),\\left(-32.98672286269284,-32.986722862692815\\right),\\left(-36.128315516282626,-36.12831551628261\\right),\\left(-39.26990816987242,-39.2699081698724\\right),\\left(-42.41150082346221,-42.4115008234622\\right),\\left(-45.55309347705201,-45.55309347705199\\right),\\left(-48.6946861306418,-48.694686130641784\\right),\\left(-51.836278784231595,-51.836278784231574\\right),\\left(-54.977871437821385,-54.97787143782137\\right),\\left(-58.11946409141118,-58.11946409141116\\right),\\left(-61.26105674500097,-61.26105674500096\\right),\\left(-64.40264939859077,-64.40264939859075\\right),\\left(-67.54424205218056,-67.54424205218054\\right),\\left(-70.68583470577035,-70.68583470577033\\right),\\left(-73.82742735936014,-73.82742735936013\\right),\\left(-76.96902001294994,-76.96902001294993\\right),\\left(-80.11061266653974,-80.11061266653971\\right),\\left(-83.25220532012952,-83.2522053201295\\right),\\left(-86.39379797371932,-86.3937979737193\\right),\\left(-89.53539062730911,-89.5353906273091\\right),\\left(-92.67698328089891,-92.67698328089888\\right),\\left(-95.81857593448869,-95.81857593448868\\right),\\left(-98.96016858807849,-98.96016858807847\\right),\\left(-102.10176124166829,-102.10176124166827\\right),\\left(-105.24335389525808,-105.24335389525805\\right),\\left(-108.38494654884788,-108.38494654884785\\right),\\left(-111.52653920243766,-111.52653920243765\\right),\\left(-114.66813185602746,-114.66813185602744\\right),\\left(-117.80972450961725,-117.80972450961724\\right),\\left(-120.95131716320705,-120.95131716320702\\right),\\left(-124.09290981679683,-124.09290981679682\\right)\\right]"},{"type":"expression","id":"18","folderId":"65","color":"#2d70b3","latex":"\\left[\\operatorname{polygon}\\left(Q\\left[2k-1\\right],\\left(Q\\left[2k-1\\right].x,Q\\left[2k\\right].y\\right),Q\\left[2k\\right],\\left(Q\\left[2k\\right].x,Q\\left[2k-1\\right].y\\right)\\right)\\operatorname{for}k=\\left[1\\right]\\right]","hidden":true,"fillOpacity":"0.5","lineOpacity":"1","lineWidth":"10"},{"type":"expression","id":"38","folderId":"65","color":"#2d70b3","latex":"\\left[\\operatorname{polygon}\\left(Q\\left[2k-1\\right],\\left(Q\\left[2k-1\\right].x,Q\\left[2k\\right].y\\right),Q\\left[2k\\right],\\left(Q\\left[2k\\right].x,Q\\left[2k-1\\right].y\\right)\\right)\\operatorname{for}k=\\left[2...\\left(\\frac{\\operatorname{length}\\left(Q\\right)}{2}\\right)\\right]\\right]","hidden":true,"fillOpacity":"0.3","lineOpacity":"0.5","lineWidth":"10"},{"type":"text","id":"61","text":"reset"},{"type":"expression","id":"89","color":"#c74440","latex":"K=39"},{"type":"expression","id":"62","color":"#000000","latex":"Q\\to\\left[\\left(-K,-K\\right),\\left(K,K\\right)\\right]+\\left(\\operatorname{random}\\left(\\right),\\operatorname{random}\\left(\\right)\\right)"},{"type":"expression","id":"103","color":"#388c46","latex":"Q\\to\\operatorname{join}\\left(\\left(1+\\left[0...K\\right]\\pi,0\\right),\\left(-\\left[0...K\\right]\\pi-1,\\left[0...K\\right]\\pi+1.7\\right),\\left(-\\left[0...K\\right]\\pi-1,-\\left[0...K\\right]\\pi-1.7\\right)\\right)"},{"type":"text","id":"50","text":"argument principle integral (zeroes - poles), in terms of box-corners"},{"type":"expression","id":"26","color":"#6042a6","latex":"I_{C}\\left(a,b\\right)=\\frac{1}{2\\pi}\\left(\\int_{a.x}^{b.x}A_{rg}\\left(\\left(t,a.y\\right)\\right).y\\ dt+\\int_{a.y}^{b.y}A_{rg}\\left(\\left(b.x,t\\right)\\right).x\\ dt-\\int_{a.x}^{b.x}A_{rg}\\left(\\left(t,b.y\\right)\\right).y\\ dt-\\int_{a.y}^{b.y}A_{rg}\\left(\\left(a.x,t\\right)\\right).x\\ dt\\right)"},{"type":"expression","id":"29","color":"#2d70b3","latex":"I_{C}\\left(Q\\left[1\\right],Q\\left[2\\right]\\right)"},{"type":"text","id":"54","text":"divide box (from corners) in half"},{"type":"expression","id":"55","color":"#6042a6","latex":"D_{b}\\left(a,b\\right)=\\left\\{\\left|b.y-a.y\\right|>\\left|b.x-a.x\\right|:\\left[a,\\left(b.x,\\frac{a.y+b.y}{2}\\right),\\left(a.x,\\frac{a.y+b.y}{2}\\right),b\\right],\\left[a,\\left(\\frac{a.x+b.x}{2},b.y\\right),\\left(\\frac{a.x+b.x}{2},a.y\\right),b\\right]\\right\\}"},{"type":"text","id":"52","text":"step in procedure (binary-search mode)"},{"type":"expression","id":"35","color":"#388c46","latex":"S_{b}=\\left\\{\\operatorname{length}\\left(Q\\right)\\ge2:\\left(Q\\to\\operatorname{join}\\left(Q\\left[3...\\right],\\left\\{I_{C}\\left(Q\\left[1\\right],Q\\left[2\\right]\\right)>\\frac{1}{2}:D_{b}\\left(Q\\left[1\\right],Q\\left[2\\right]\\right),\\left[\\right]\\right\\}\\right)\\right)\\right\\}"},{"type":"text","id":"87","text":"round to 13 significant figures"},{"type":"expression","id":"83","color":"#000000","latex":"R_{s}\\left(x\\right)=\\operatorname{sgn}\\left(x\\right)10^{\\operatorname{floor}\\left(\\log_{10}\\left(\\left|x\\right|\\right)\\right)-13}\\operatorname{round}\\left(10^{13-\\operatorname{floor}\\left(\\log_{10}\\left(\\left|x\\right|\\right)\\right)}\\left|x\\right|\\right)","hidden":true},{"type":"expression","id":"88","color":"#6042a6","latex":"R_{v}\\left(p\\right)=\\left(R_{s}\\left(p.x\\right),R_{s}\\left(p.y\\right)\\right)"},{"type":"text","id":"79","text":"step in procedure (Newton's-method mode)"},{"type":"expression","id":"80","color":"#2d70b3","latex":"S_{n}=\\left(Q\\to\\operatorname{unique}\\left(Q-M_{c}\\left(F\\left(Q\\right),R_{c}\\left(G\\left(Q\\right)\\right)\\right)\\right)\\right)"},{"type":"expression","id":"82","color":"#6042a6","latex":"\\operatorname{length}\\left(Q\\right)"},{"type":"expression","id":"118","color":"#2d70b3","latex":"L=Q\\left[\\operatorname{distance}\\left(Q,\\left(0,0\\right)\\right)<150\\right]"},{"type":"expression","id":"115","color":"#6042a6","latex":"\\operatorname{mean}\\left(L.x\\right)"},{"type":"expression","id":"116","color":"#000000","latex":"\\operatorname{mean}\\left(L.y\\right)"},{"type":"text","id":"120","text":"e^z + cos z -> mean 0"},{"type":"text","id":"122","text":"e^z + sin z -> mean 0"},{"type":"text","id":"124","text":"2 e^z cos z + 1 -> mean 0"},{"type":"expression","id":"91","color":"#c74440","latex":"\\sin x+e^{x}=0","hidden":true},{"type":"expression","id":"92","color":"#2d70b3","latex":"y=\\left[1,-1\\right]x","hidden":true}],"ticker":{"handlerLatex":"S_{b}","minStepLatex":"100","open":true}}}