2d:null 2e:["$","title",null,{"children":"Complex | Vinícius Mello | Observable"}] 33:T489b,ℂ = { const re = (z) => (z.constructor === Array) ? z[0] : z; const im = (z) => (z.constructor === Array) ? z[1] : 0; const to = (z) => (z.constructor === Array) ? z : [z,0]; const I = [0,1]; const add2 = (z,w) => [re(z)+re(w), im(z)+im(w)]; const add = (...z) => { let s = z[0]; for(let i=1;i [re(z)-re(w), im(z)-im(w)]; const neg = (z) => [-re(z), -im(z)]; const mul2 = (z,w) => [re(z)*re(w)-im(z)*im(w), re(z)*im(w)+im(z)*re(w)]; const mul = (...z) => { let p = z[0]; for(let i=1;i [re(z), -im(z)]; const abs2 = (z) => re(z)*re(z)+im(z)*im(z); const abs = (z) => Math.sqrt(abs2(z)); const div = (z,w) => mul(z,conj(w),1/abs2(w)); const arg = z => Math.atan2(im(z), re(z)); let zeroTol = 1e-10; const setZeroTol = (tol) => { zeroTol = tol; } const isZero = (r) => Math.abs(r) { let d = sub(z,w); return isZero(re(d)) && isZero(im(d)); } const chop = (z) => { let r = isZero(re(z)) ? 0 : re(z); let i = isZero(im(z)) ? 0 : im(z); if(i==0) return r; return [r,i]; } const exp = (z) => { let r = Math.exp(re(z)); let th = im(z); return [r*Math.cos(th),r*Math.sin(th)]; } const ln = z => { if(isZero(im(z)) && re(z)>0) return Math.log(re(z)); return [Math.log(abs(z)), arg(z)]; } const pow = (base, power) => { if(equal(base,[0,0])) return 0; let r = exp(mul(power, ln(base))); return r; } const sqrt = z => { if(equal(z,[0,0])) return 0; return pow(z,0.5); } const cos = z => { if(isZero(im(z))) return Math.cos(re(z)); const iz = mul(I, z); return div(add2(exp(iz), exp(neg(iz))),2); }; const cosh = z => { if(isZero(im(z))) return Math.cosh(re(z)); return div(add2(exp(z), exp(neg(z))), 2); } const sin = z => { if(isZero(im(z))) return Math.sin(re(z)); const iz = mul(I, z); return div(sub(exp(iz), exp(neg(iz))),[0,2]); }; const sinh = z => { if(isZero(im(z))) return Math.sinh(re(z)); return div(sub(exp(z), exp(neg(z))), 2); } const acos = z => { if(isZero(im(z)) && Math.abs(re(z))<=1 ) return Math.acos(re(z)); return neg(mul2(I,ln(add2(z,sqrt(sub(mul2(z,z),1)))))); } //adapted from https://github.com/paulmasson/math/tree/master/src /* MIT License Copyright © 2017-2019 Paul Masson Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: */ const carlsonRC = (x,y) => { if ( equal(x,y) ) return div( 1, sqrt(x) ); return div( acos( sqrt( div(x,y) ) ), sqrt( sub(y,x) ) ); } const carlsonRF = (x,y,z) => { if ( equal(x, y) ) return carlsonRC(z,x); if ( equal(x, z) ) return carlsonRC(y,x); if ( equal(y, z) ) return carlsonRC(x,y); let xm = x; let ym = y; let zm = z; let A0 = div( add( x, y, z ), 3 ); let Am = [re(A0), im(A0)]; let Q = Math.pow( 3*zeroTol, -1/6 ) * Math.max( abs( sub(A0,x) ), abs( sub(A0,y) ), abs( sub(A0,z) ) ); let g = .25; let pow4 = 1; let m = 0; while ( m<5000 ) { let xs = sqrt(xm); let ys = sqrt(ym); let zs = sqrt(zm); let lm = add( add(mul2(xs,ys), mul2(xs,zs)), mul2(ys,zs) ); let Am1 = mul2( add2(Am,lm), g ); xm = mul2( add2(xm,lm), g ); ym = mul2( add2(ym,lm), g ); zm = mul2( add2(zm,lm), g ); if ( pow4 * Q < abs(Am) ) break; Am = Am1; m += 1; pow4 *= g; } let t = div( pow4, Am ); let X = mul2( sub(A0,x), t ); let Y = mul2( sub(A0,y), t ); let Z = neg( add(X,Y) ); let E2 = sub( mul2(X,Y), mul2(Z,Z) ); let E3 = mul(X,Y,Z); return mul( pow( Am, -0.5 ), add( 9240, mul2(-924,E2), mul(385,E2,E2), mul2(660,E3), mul(-630,E2,E3)), 1/9240 ); } const ellipticK = (m) => carlsonRF(0,sub(1,m),1); const ellipticF = (x,m) => { let period; if ( Math.abs(re(x)) > Math.PI / 2 ) { let p = Math.round( re(x) / Math.PI ); x = [re(x) - p * Math.PI, im(x)]; period = mul2( 2 * p, ellipticK( m ) ); } return add2( mul2( sin(x), carlsonRF( mul2(cos(x),cos(x)), sub( 1, mul(m,sin(x),sin(x)) ), 1) ), period ); } const ellipticNome = (m) => { return exp( div( mul2( -Math.PI, ellipticK( sub(1,m) ) ), ellipticK(m) ) ); } const theta1 = (z,q) => { let p = 1; let s = 0; let i = 0; let sgn = 1; while ( Math.abs(re(p)) > zeroTol || Math.abs(im(p)) > zeroTol ) { p = mul( sgn, pow( q, i*i+i ), sin( mul2(2*i+1,z) ) ); s = add2( s, p ); i++; sgn *= -1; } return mul( 2, pow( q, 1/4 ), s ); } const theta2 = (z,q) => { let p = 1; let s = 0; let i = 0; while ( Math.abs(re(p)) > zeroTol || Math.abs(im(p)) > zeroTol ) { p = mul2( pow( q, i*i+i ), cos( mul2(2*i+1,z) ) ); s = add2( s, p ); i++; } return mul( 2, pow( q, 1/4 ), s ); } const theta3 = (z,q) => { let p = 1; let s = 0; let i = 1; while ( Math.abs(re(p)) > zeroTol || Math.abs(im(p)) > zeroTol ) { p = mul2( pow( q, i*i ), cos( mul2(2*i,z) ) ); s = add2( s, p ); i++; } return add2( 1, mul2(2,s) ); } const theta4 = (z,q) => { let p = 1; let s = 0; let i = 1; while ( Math.abs(re(p)) > zeroTol || Math.abs(im(p)) > zeroTol ) { p = mul2( pow( neg(q), i*i ), cos( mul2(2*i,z) ) ); s = add2( s, p ); i++; } return add2( 1, mul2(2,s) ); } const sn = (z,m) => { let q = ellipticNome(m); let th3 = theta3(0,q); let t = div( z, mul2( th3, th3 ) ); return mul2( div( th3, theta2(0,q) ), div( theta1(t,q), theta4(t,q) ) ); } const cubicTrigSolution = ( p, q, n ) => { // p, q both negative in defining cubic if(equal(p,[0,0])) { let u3 = [-0.5,-Math.sqrt(3)/2]; return mul2(pow(q,1/3),pow(u3,n)); } return mul( 2/Math.sqrt(3), sqrt(p), cos( sub( div( acos( mul( 3*Math.sqrt(3)/2, q, pow(p,-3/2) ) ), 3 ), 2*Math.PI*n/3 ) ) ); } let wPData = {}; const setWeierstrassPRoots = (g2,g3) => { g2 = div( g2, 4 ); g3 = div( g3, 4 ); wPData.e1 = cubicTrigSolution( g2, g3, 0 ); wPData.e2 = cubicTrigSolution( g2, g3, 1 ); wPData.e3 = cubicTrigSolution( g2, g3, 2 ); wPData.e13 = sub(wPData.e1, wPData.e3); wPData.e23 = sub(wPData.e2, wPData.e3); wPData.m = div( wPData.e23, wPData.e13 ); } const weierstrassP = (z) => { // Whittaker & Watson, Section 22.351 return add( wPData.e3, div( wPData.e13, pow( sn( mul2( z, sqrt(wPData.e13) ), wPData.m ), 2 ) ) ); } setWeierstrassPRoots(4,0); const logGamma = (z) => { let c = [ 57.1562356658629235, -59.5979603554754912, 14.1360979747417471, -0.491913816097620199, .339946499848118887e-4, .465236289270485756e-4, -.983744753048795646e-4, .158088703224912494e-3, -.210264441724104883e-3, .217439618115212643e-3, -.164318106536763890e-3, .844182239838527433e-4, -.261908384015814087e-4, .368991826595316234e-5 ]; if ( Number.isInteger(re(z)) && re(z) <= 0 && im(z) === 0 ) return Infinity; // reflection formula with modified Hare correction to imaginary part if ( re(z) < 0 ) { let t = sub( ln( div( Math.PI, sin( mul2(Math.PI,z) ) ) ), logGamma( sub(1,z) ) ); let s = im(z) < 0 ? -1 : 1; let d = im(z) === 0 ? 1/4 : 0; let k = Math.ceil( re(z)/2 - 3/4 + d ); return add2( t, [ 0, 2*s*k*Math.PI ] ); } let t = add2( z, 5.24218750000000000 ); t = sub( mul2( add2( z, 0.5 ), ln(t)), t ); let s = 0.999999999999997092; for ( let j = 0 ; j < 14 ; j++ ) s = add2( s, div( c[j], add2( z, j+1 ) ) ); let u = add2( t, ln( mul2( 2.5066282746310005, div( s, z ) ) ) ); // adjustment to keep imaginary part on same sheet if ( re(s) < 0 ) { if( im(z) < 0 && im(div(s,z)) < 0 ) u = add2( u, [0,2*Math.PI] ); if( im(z) > 0 && im(div(s,z)) > 0 ) u = add2( u, [0,-2*Math.PI] ); } return u; } const gamma = (z) => exp(logGamma(z)); const factorials = [ 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 0.243290200817664e19, 0.5109094217170944e20, 0.112400072777760768e22, 0.2585201673888497664e23, 0.62044840173323943936e24, 0.15511210043330985984e26, 0.403291461126605635584e27, 0.10888869450418352160768e29, 0.304888344611713860501504e30, 0.8841761993739701954543616e31, 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0.308976961384735088795856467036324046592e272, 0.4789142901463393876335775239063022722176e274, 0.7471062926282894447083809372938315446595e276, 0.1172956879426414428192158071551315525115e279, 0.1853271869493734796543609753051078529682e281, 0.2946702272495038326504339507351214862195e283, 0.4714723635992061322406943211761943779512e285, 0.7590705053947218729075178570936729485014e287, 0.1229694218739449434110178928491750176572e290, 0.2004401576545302577599591653441552787813e292, 0.3287218585534296227263330311644146572013e294, 0.5423910666131588774984495014212841843822e296, 0.9003691705778437366474261723593317460744e298, 0.1503616514864999040201201707840084015944e301, 0.2526075744973198387538018869171341146786e303, 0.4269068009004705274939251888899566538069e305, 0.7257415615307998967396728211129263114717e307 ]; const factorial = (n) => factorials[n]; const zeta = (z) => { // functional equation if ( re(z) < 0 ) return mul( pow(2,z), pow(Math.PI,sub(z,1)), sin( mul2(Math.PI/2,z) ), gamma( sub(1,z) ), zeta( sub(1,z) ) ); // Borwein algorithm let n = 14; // from error bound for tolerance if ( !isZero(im(z)) ) n = Math.max( n, Math.ceil( ln( 2 / abs(gamma(z)) / zeroTol ) / ln( 3 + Math.sqrt(8) ) ) ); let d = [ 1 ]; for ( let i = 1 ; i <= n ; i++ ) // order of multiplication reduces overflow, but factorial overflows at 171 d.push( d[i-1] + n * factorial( n+i-1 ) / factorial( n-i ) / factorial( 2*i ) * 4**i ); let s = 0; for ( let k = 0 ; k < n ; k++ ) s = add( s, div( (-1)**k * ( d[k] - d[n] ), pow( k+1, z ) ) ); return div( div( s, -d[n] ), sub( 1, pow( 2, sub(1,z) ) ) ); } // https://en.wikipedia.org/wiki/Durand%E2%80%93Kerner_method const roots = (p, deg, R=1) => { function iterate(zs) { const zero = [0,0]; const niter = 100*deg; for(let it=0; it '+parse(f));\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":307,"value":"function fzw(f) {\n let c = ℂ;\n //let fw = eval('(z,w) => '+parse(f));\n //return (z) => (w) => fw(z,w);\n return eval('(z) => (w) => '+parse(f)); \n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":254,"value":"mathjs = require('mathjs')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W34","stdlib":"1","titling":"auto"}} 32:"$31:body"