--- title: "Beautiful complex functions" author: "Stéphane Laurent" date: '2022-06-20' tags: R, maths, graphics, special-functions rbloggers: yes output: md_document: variant: markdown preserve_yaml: true html_document: highlight: kate keep_md: no highlighter: pandoc-solarized --- I've just submitted a new package to CRAN: **jacobi**. It allows to evaluate the Jacobi theta functions as well as some related functions. In particular, some Eisenstein series, that you will see below. To represent a complex function, a color map is needed, that is to say a function which maps each complex number to a color. I'm using two different color maps: ``` {.r .numberLines} modulo <- function(a, p) { a - p * ifelse(a > 0, floor(a/p), ceiling(a/p)) } colormap1 <- function(z){ if(is.infinite(z) || is.nan(z)) return("#000000") x <- Re(z) y <- Im(z) a <- atan2(y, x) r <- modulo(Mod(z), 1) g <- abs(modulo(a, 0.5)) * 2 b <- abs(modulo(x*y, 1)) if(is.nan(b)){ return("#000000") } rgb( (1.0 - cos(r-0.5))*8.0, (1.0 - cos(g-0.5))*8.0, (1.0 - cos(b-0.5))*8.0, maxColorValue = 1 ) } colormap2 <- function(z){ if(is.infinite(z) || is.nan(z)) return("#000000") arg <- Arg(z) if(arg < 0) arg <- 2*pi + arg h <- arg / 2 / pi s <- sqrt((1 + sin(2*pi*log(1+Mod(z))))/2) v <- (1 + cos(2*pi*log(1+Mod(z))))/2 hsv(h, s, v) } ``` Here is the \$E_4\$ Eisenstein series: ``` {.r .numberLines} # background color bkgcol <- rgb(21, 25, 30, maxColorValue = 255) f <- Vectorize(function(x, y){ q <- x + 1i*y if(Mod(q) >= 0.99 || (Im(q) == 0 && Re(q) <= 0)) return(bkgcol) z <- En(4, q) colormap1(1/z) }) x <- y <- seq(-1, 1, len = 2000) image <- outer(x, y, f) opar <- par(mar = c(0,0,0,0), bg = bkgcol) plot( c(-100, 100), c(-100, 100), type = "n", xlab = "", ylab = "", axes = FALSE, asp = 1 ) rasterImage(image, -100, -100, 100, 100) par(opar) ``` ![](./figures/Eisenstein4.png){width="55%"} It makes me think to a beetle. And here is the \$E_6\$ Eisenstein series: ``` {.r .numberLines} f <- Vectorize(function(x, y){ q <- x + 1i*y if(Mod(q) >= 0.99 || (Im(q) == 0 && Re(q) <= 0)) return(bkgcol) z <- En(6, q) colormap2(z) }) ``` ![](./figures/Eisenstein6.png){width="55%"}