> x <- seq(0.01, 1, 0.001)
> z <- data.frame(matrix(data = NA, ncol = 10, nrow = length(x)))
> colnames(z) <- c("x", paste("n", 1:9, sep = ""))
> z$x <- x
> i <- 1
> for (i in 1:9) {
+     k <- (i)
+     condition <- z$x > 0 & z$x < 1/k^2
+     z[condition, (i + 1)] <- 0
+     z[!condition, (i + 1)] <- 1/sqrt(z[!condition, 1])
+ }
> plot.new()
> cols <- rainbow(9)
> for (j in 2:10) {
+     par(new = T)
+     plot(z$x, z[, j], col = cols[(j - 1)], type = "l", ylim = c(0,
+         10))
+ }
> par(new = T)
> plot(z$x, z$x^(-0.5), col = "black", type = "l", ylim = c(0,
+     10), lwd = 2)

Approx_by_meas_funcs-001.jpg

> plot(z$x, (1 - z$x^2)^(-0.5), col = "black", type = "l", ylim = c(0,
+     10), lwd = 2)

Approx_by_meas_funcs-002.jpg

> plot(seq(-120, 120, 1), atan(seq(-120, 120, 1)))

Approx_by_meas_funcs-003.jpg