{"version":9,"randomSeed":"63ac44b2296c44c6423ab740c0c0cc69","graph":{"viewport":{"xmin":-10,"ymin":-6.8506998444790055,"xmax":9.999999999999996,"ymax":6.850699844479002}},"expressions":{"list":[{"type":"folder","id":"11","title":"complex function domain colouring library"},{"type":"expression","id":"69","folderId":"11","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":"70","folderId":"11","color":"#c74440","latex":"A_{rg}\\left(z\\right)=\\arctan\\left(z.y,z.x\\right)"},{"type":"expression","id":"71","folderId":"11","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|\\le\\frac{\\pi}{N_{cps}}","lines":true,"colorLatex":"C_{cp}","fillOpacity":"1"},{"type":"expression","id":"72","folderId":"11","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]}","lines":false,"labelSize":"medium","fillOpacity":"\\frac{1}{N_{cps}}"},{"type":"expression","id":"73","folderId":"11","color":"#388c46","latex":"N_{cps}=8","slider":{"hardMin":true,"hardMax":true,"min":"1","max":"100","step":"1"}},{"type":"folder","id":"32","title":"basic functions of complex numbers","collapsed":true},{"type":"text","id":"33","folderId":"32","text":"multiply"},{"type":"expression","id":"34","folderId":"32","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":"35","folderId":"32","text":"reciprocate"},{"type":"expression","id":"36","folderId":"32","color":"#388c46","latex":"R_{c}\\left(z\\right)=\\frac{\\left(z.x,-z.y\\right)}{z.x^{2}+z.y^{2}}"},{"type":"text","id":"37","folderId":"32","text":"square"},{"type":"expression","id":"38","folderId":"32","color":"#2d70b3","latex":"Q_{c}\\left(z\\right)=M_{c}\\left(z,z\\right)"},{"type":"text","id":"39","folderId":"32","text":"cube"},{"type":"expression","id":"40","folderId":"32","color":"#000000","latex":"C_{c}\\left(z\\right)=M_{c}\\left(Q_{c}\\left(z\\right),z\\right)"},{"type":"text","id":"41","folderId":"32","text":"naturally exponentiate"},{"type":"expression","id":"42","folderId":"32","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":"43","folderId":"32","text":"naturally logarithmise"},{"type":"expression","id":"44","folderId":"32","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":"27","title":"gamma function implementation"},{"type":"text","id":"51","folderId":"27","text":"https://mathworld.wolfram.com/WeierstrassForm.html"},{"type":"expression","id":"47","folderId":"27","color":"#2d70b3","latex":"\\gamma=0.577215664901532"},{"type":"expression","id":"48","folderId":"27","color":"#c74440","latex":"W_{T}\\left(z,r\\right)=M_{c}\\left(\\left(\\left(1,0\\right)+\\frac{z}{r}\\right),E_{c}\\left(-\\frac{z}{r}\\right)\\right)"},{"type":"expression","id":"49","folderId":"27","color":"#2d70b3","latex":"W_{T4}\\left(z,r\\right)=M_{c}\\left(M_{c}\\left(W_{T}\\left(z,r\\right),W_{T}\\left(z,r+1\\right)\\right),M_{c}\\left(W_{T}\\left(z,r+2\\right),W_{T}\\left(z,r+3\\right)\\right)\\right)"},{"type":"expression","id":"30","folderId":"27","color":"#000000","latex":"\\Gamma_{12}\\left(z\\right)=R_{c}\\left(M_{c}\\left(M_{c}\\left(M_{c}\\left(z,E_{c}\\left(\\gamma z\\right)\\right),W_{T4}\\left(z,1\\right)\\right),M_{c}\\left(W_{T4}\\left(z,5\\right),W_{T4}\\left(z,9\\right)\\right)\\right)\\right)"},{"type":"text","id":"62","folderId":"27","text":"the below methods converge well, but are very bad for values near the negative reals"},{"type":"text","id":"55","folderId":"27","text":"https://mathworld.wolfram.com/StirlingsApproximation.html"},{"type":"expression","id":"56","folderId":"27","color":"#2d70b3","latex":"\\Gamma_{S}\\left(z\\right)=\\sqrt{2\\pi}\\cdot E_{c}\\left(M_{c}\\left(z+\\left(\\frac{1}{2},0\\right),L_{c}\\left(z\\right)\\right)-z\\right)","labelSize":"medium"},{"type":"text","id":"59","folderId":"27","text":"https://mathworld.wolfram.com/StirlingsSeries.html"},{"type":"expression","id":"60","folderId":"27","color":"#000000","latex":"\\Gamma_{S2}\\left(z\\right)=\\sqrt{2\\pi}\\cdot M_{c}\\left(E_{c}\\left(M_{c}\\left(z-\\left(\\frac{1}{2},0\\right),L_{c}\\left(z\\right)\\right)-z\\right),\\left(\\left(1,0\\right)+\\frac{1}{12}R_{c}\\left(z\\right)\\right)\\right)"},{"type":"text","id":"64","folderId":"27","text":"https://en.wikipedia.org/wiki/Stirling%27s_approximation#A_convergent_version_of_Stirling%27s_formula"},{"type":"expression","id":"67","folderId":"27","color":"#6042a6","latex":"\\Gamma_{SC3}\\left(z\\right)=E_{c}\\left(M_{c}\\left(z,L_{c}\\left(z\\right)\\right)-z+\\frac{1}{2}L_{c}\\left(2\\pi R_{c}\\left(z\\right)\\right)+\\frac{1}{12}R_{c}\\left(z+\\left(1,0\\right)\\right)+\\frac{1}{12}R_{c}\\left(M_{c}\\left(z+\\left(1,0\\right),z+\\left(2,0\\right)\\right)\\right)\\right)"},{"type":"text","id":"75","folderId":"27","text":"what if we combined the methods?"},{"type":"expression","id":"76","folderId":"27","color":"#2d70b3","latex":"\\Gamma_{M}\\left(z\\right)=\\left\\{z.x<0:\\Gamma_{12}\\left(z\\right),\\Gamma_{SC3}\\left(z\\right)\\right\\}"},{"type":"expression","id":"22","color":"#000000","latex":"f\\left(z\\right)=\\left(z.x\\cdot z.x-z.y\\cdot z.y+1,z.x\\cdot z.y\\right)"},{"type":"expression","id":"23","color":"#c74440","latex":"g\\left(z\\right)=z.x^{2}+z.y^{2}"},{"type":"expression","id":"17","color":"#6042a6","latex":"F_{comp}\\left(z\\right)=\\Gamma_{M}\\left(z\\right)","hidden":true},{"type":"expression","id":"68","color":"#c74440","latex":"\\Gamma_{S2}\\left(\\left(0,-\\frac{3}{2}\\right)\\right).x","labelSize":"medium"}]}}