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。
図１: グーデルマン関数 gd(x)、誤差関数 erf(x)、tanh(x)、ランジュバン関数 L(x)
Wikipedia→グーデルマン関数
地球上のある地点の緯度をφとすると、メルカトル図法による地図上でその地点の赤道からの緯線距離はグーデルマン関数の逆関数gd-1(φ)に比例する。
図２: デュースブルク−エッセン大学（旧メルカトル大学）の物理学科の建物 Keksdosen 。
Wikipedia→誤差関数 解析的にフーリエ変換できる。
Wikipedia→tanh
Wikipedia→ランジュバン関数
二重指数関数型数値積分公式に使われる。
図１のEPSファイルを出力するGNUPLOTスクリプトが下のGudermannian-erf-tanh-Langevin1.gp。
| #!/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: |
| /* -*-CSS-*- | |
| * style.css for README.html of feram | |
| * Time-stamp: <2014-07-22 10:47:37 takeshi> | |
| * Author: Takeshi NISHIMATSU | |
| */ | |
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| font-size: larger; | |
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| white-space: pre; | |
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| } | |
| p img { | |
| width: 60%; | |
| margin: auto; | |
| display: block; | |
| } | |
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| width: 60%; | |
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| margin-left: 10%; | |
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