{"version":9,"randomSeed":"478b19383896d33640f8c4ba66bc1732","graph":{"viewport":{"xmin":-19.999999999999996,"ymin":-17.150069507214255,"xmax":20.000000000000004,"ymax":17.150069507214248}},"expressions":{"list":[{"type":"expression","id":"3","color":"#388c46","latex":"g\\left(z\\right)=Q_{c}\\left(z\\right)+C_{c}\\left(z\\right)+\\left(7,3\\right)+4\\cdot z","labelSize":"medium"},{"type":"expression","id":"30","color":"#6042a6","latex":"f\\left(x,y\\right)=g\\left(\\left(x,y\\right)\\right)","labelSize":"medium"},{"type":"text","id":"34","text":"https://mathworld.wolfram.com/Cauchy-RiemannEquations.html"},{"type":"expression","id":"31","color":"#000000","latex":"g_{d}\\left(x,y\\right)=\\frac{1}{2}\\cdot\\left(\\frac{d}{dx}f\\left(x,y\\right)+M_{c}\\left(\\left(0,-1\\right),\\frac{d}{dy}f\\left(x,y\\right)\\right)\\right)"},{"type":"expression","id":"38","color":"#c74440","latex":"N_{g}\\left(z\\right)=z-M_{c}\\left(g\\left(z\\right),R_{c}\\left(g_{d}\\left(z.x,z.y\\right)\\right)\\right)","labelSize":"medium"},{"type":"folder","id":"14","title":"basic functions of complex numbers","collapsed":true},{"type":"text","id":"15","folderId":"14","text":"multiply"},{"type":"expression","id":"16","folderId":"14","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":"17","folderId":"14","text":"reciprocate"},{"type":"expression","id":"18","folderId":"14","color":"#388c46","latex":"R_{c}\\left(z\\right)=\\frac{\\left(z.x,-z.y\\right)}{z.x^{2}+z.y^{2}}"},{"type":"text","id":"19","folderId":"14","text":"square"},{"type":"expression","id":"20","folderId":"14","color":"#2d70b3","latex":"Q_{c}\\left(z\\right)=M_{c}\\left(z,z\\right)"},{"type":"text","id":"21","folderId":"14","text":"cube"},{"type":"expression","id":"22","folderId":"14","color":"#000000","latex":"C_{c}\\left(z\\right)=M_{c}\\left(Q_{c}\\left(z\\right),z\\right)"},{"type":"text","id":"23","folderId":"14","text":"naturally exponentiate"},{"type":"expression","id":"24","folderId":"14","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":"25","folderId":"14","text":"naturally logarithmise"},{"type":"expression","id":"26","folderId":"14","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":"folder","id":"6","title":"complex function domain colouring library"},{"type":"expression","id":"7","folderId":"6","color":"#2d70b3","latex":"C_{cp}=\\operatorname{hsv}\\left(\\operatorname{mod}\\left(\\left[0,\\frac{360}{N_{cps}},...360\\right]-60,360\\right),1,1\\right)"},{"type":"expression","id":"8","folderId":"6","color":"#c74440","latex":"A_{rg}\\left(z\\right)=\\arctan\\left(z.y,z.x\\right)"},{"type":"expression","id":"9","folderId":"6","color":"#2d70b3","latex":"\\left|A_{rg}\\left(F_{comp}\\left(\\left(x,y\\right)\\right)\\right)-\\frac{2\\pi}{N_{cps}}\\cdot\\left[-\\frac{N_{cps}}{2},-\\frac{N_{cps}}{2}+1,...\\frac{N_{cps}}{2}\\right]\\right|=\\frac{\\pi}{N_{cps}}","lines":true,"colorLatex":"C_{cp}","fillOpacity":"1"},{"type":"expression","id":"10","folderId":"6","color":"#000000","latex":"\\operatorname{distance}\\left(\\left(0,0\\right),F_{comp}\\left(\\left(x,y\\right)\\right)\\right)\\le e^{5\\cdot\\left[-1,-1+\\frac{1}{N_{cps}},...1\\right]}","hidden":true,"lines":false,"labelSize":"medium","fillOpacity":"\\frac{2}{N_{cps}}"},{"type":"expression","id":"11","folderId":"6","color":"#388c46","latex":"N_{cps}=8","slider":{"hardMin":true,"hardMax":true,"min":"1","max":"100","step":"1"}},{"type":"expression","id":"12","folderId":"6","color":"#6042a6","latex":"F_{comp}\\left(z\\right)=g\\left(z\\right)","hidden":true},{"type":"expression","id":"36","color":"#6042a6","latex":"W=\\operatorname{rgb}\\left(255,255,255\\right)","labelSize":"medium"},{"type":"expression","id":"35","color":"#000000","latex":"z_{0}=\\left(7.7,2.97\\right)"},{"type":"folder","id":"47","title":"Newton's method steps","hidden":true,"collapsed":true},{"type":"expression","id":"37","folderId":"47","color":"#000000","latex":"z_{1}=N_{g}\\left(z_{0}\\right)","colorLatex":"W"},{"type":"expression","id":"40","folderId":"47","color":"#000000","latex":"z_{2}=N_{g}\\left(z_{1}\\right)","colorLatex":"W"},{"type":"expression","id":"41","folderId":"47","color":"#000000","latex":"z_{3}=N_{g}\\left(z_{2}\\right)","colorLatex":"W"},{"type":"expression","id":"42","folderId":"47","color":"#000000","latex":"z_{4}=N_{g}\\left(z_{3}\\right)","colorLatex":"W"},{"type":"expression","id":"43","folderId":"47","color":"#000000","latex":"z_{5}=N_{g}\\left(z_{4}\\right)","colorLatex":"W"},{"type":"expression","id":"50","folderId":"47","color":"#000000","latex":"z_{6}=N_{g}\\left(z_{5}\\right)","colorLatex":"W"},{"type":"expression","id":"51","folderId":"47","color":"#000000","latex":"z_{7}=N_{g}\\left(z_{6}\\right)","colorLatex":"W"},{"type":"expression","id":"52","folderId":"47","color":"#000000","latex":"z_{8}=N_{g}\\left(z_{7}\\right)","colorLatex":"W"},{"type":"expression","id":"53","folderId":"47","color":"#000000","latex":"z_{9}=N_{g}\\left(z_{8}\\right)","colorLatex":"W"},{"type":"expression","id":"44","color":"#000000","latex":"\\left[z_{0},z_{1},z_{2},z_{3},z_{4},z_{5},z_{6},z_{7},z_{8},z_{9}\\right]","lines":true,"labelSize":"medium"},{"type":"expression","id":"45","color":"#6042a6","latex":"\\operatorname{distance}\\left(\\left(0,0\\right),g\\left(z_{9}\\right)\\right)","labelSize":"medium"}]}}