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ゼロ近傍で傾きを持ち、±∞で±１か±π／２になる関数

# ゼロ近傍で傾きを持ち、±∞で±１か±π／２になる関数

ゼロ近傍で傾きを持ち、x→±∞ で f(x) → ±１ or ±π／２ になる関数、ようするに図１の５つの関数を集めてみた。

## グーデルマン関数

Wikipedia→グーデルマン関数

## 誤差関数

Wikipedia→誤差関数 解析的にフーリエ変換できる。

Wikipedia→tanh

## ランジュバン関数

Wikipedia→ランジュバン関数

## GNUPLOTスクリプト

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 #!/usr/bin/env gnuplot # Gudermannian-erf-tanh-Langevin1.gp # Gudermannian function, error function, tanh and Langevin function # Author: Takeshi Nishimatsu # Gist: https://gist.github.com/t-nissie/9579139 ## set terminal postscript eps enhanced color dashed "Times-Roman" 11.5 set size 0.5,0.5 set encoding iso_8859_1 # for longer minus set output "Gudermannian-erf-tanh-Langevin1.eps" set key at -0.01,1.4 spacing 1.4 font "Times-Roman,10.5" set grid set xlabel '{/Times-Italic x}' set ytics ("-{/Symbol p}/2" -pi/2, "-1" -1, "0" 0, "1" 1, "-{/Symbol p}/2" pi/2) plot [-6:6] [-pi/2:pi/2]\ asin(tanh(x)) t 'gd({/Times-Italic x})=sin^{-1}(tanh({/Times-Italic x}))' lw 4,\ erf(x) t 'erf({/Times-Italic x})' lw 4,\ tanh(x) t 'tanh({/Times-Italic x})' lw 4,\ 1/tanh(x)-1/x t '{/Times-Italic L}({/Times-Italic x})=coth({/Times-Italic x}) - 1/{/Times-Italic x}' lw 4,\ tanh(pi/2*sinh(x)) t '{/Symbol f}({/Times-Italic x})=tanh({/Symbol (p/2)}shinh {/Times-Italic x})' lw 4 set output !epstopdf Gudermannian-erf-tanh-Langevin1.eps !convert -density 400 Gudermannian-erf-tanh-Langevin1.eps Gudermannian-erf-tanh-Langevin1.png #Local variables: # compile-command: "gnuplot Gudermannian-erf-tanh-Langevin1.gp" #End:
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