数学 - Python - JavaScript - 解析学 - 各種の初等関数 - 三角関数(続き)、逆三角関数(双曲線関数(sinh, cosh, tanh))

解析入門〈1〉

学習環境

• Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
• Windows 10 Pro (OS)
• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Internet Explorer, Google Chrome...)用JavaScript Library: MathJax
• 参考書籍
  • JavaScriptリファレンス 第6版
  • インタラクティブ・データビジュアライゼーション

解析入門〈1〉(松坂 和夫(著)、岩波書店)の第5章(各種の初等関数)、5.4(三角関数(続き)、逆三角関数)、問題14.を取り組んでみる。

1. 
   $\begin{array}{l}\frac{e^{2x}+e^{-2x}+2}{4}-\frac{e^{2x}+e^{-2x}-2}{4}=1\\ \frac{4}{e^{2x}+e^{-2x}+2}\\ =\frac{e^{2x}+e^{-2x}+2-\left(e^{2x}+e^{-2x}+2\right)+4}{e^{2x}+e^{-2x}+2}\\ =1-\frac{e^{2x}+e^{-2x}-2}{e^{2x}+e^{-2x}+2}\\ =1-\frac{{\left(e^x-e^{-x}\right)}^2}{{\left(e^x+e^{-x}\right)}^2}\\ =1-{\tanh }^{2}x\end{array}$
2. 
   $\begin{array}{l}\frac{e^x-e^{-x}}{2}·\frac{e^y+e^{-y}}{2}+\frac{e^x+e^{-x}}{2}·\frac{e^y-e^{-y}}{2}\\ =\frac{e^{x+y}-e^{-\left(x+y\right)}}{2}\end{array}$
3. 
   $\begin{array}{l}\frac{e^x+e^{-x}}{2}·\frac{e^y+e^{-y}}{2}+\frac{e^x-e^{-x}}{2}·\frac{e^y-e^{-y}}{2}\\ =\frac{e^{x+y}+e^{-\left(x+y\right)}}{2}\end{array}$
4. 
   $\begin{array}{l}\frac{e^x+e^{-x}}{2}\\ \frac{e^x-e^{-x}}{2}\\ \frac{{\left(e^x+e^{-x}\right)}^2-{\left(e^x-e^{-x}\right)}^2}{{\left(e^x+e^{-x}\right)}^2}\\ =\frac{4}{{\left(e^x+e^{-x}\right)}^2}\\ =\frac{1}{{\left(\frac{e^x+e^{-x}}{2}\right)}^2}\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from sympy import symbols, sinh, cosh, tanh, pprint, Derivative

print('d.')
x = symbols('x')
for i, expr in enumerate([sinh(x), cosh(x), tanh(x)], 1):
    print(i)
    pprint(expr)
    d = Derivative(expr, x)
    pprint(d)
    pprint(d.doit())

入出力結果(Terminal, IPython)

$ ./sample14.py
d.
1
sinh(x)
d          
──(sinh(x))
dx         
cosh(x)
2
cosh(x)
d          
──(cosh(x))
dx         
sinh(x)
3
tanh(x)
d          
──(tanh(x))
dx         
      2       
- tanh (x) + 1
$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="0.5">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.0001" value="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5">

<button id="draw0">draw</button>
<button id="clear0">clear</button>

<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>

<script src="sample14.js"></script>    

JavaScript

let div0 = document.querySelector('#graph0'),
    pre0 = document.querySelector('#output0'),
    width = 600,
    height = 600,
    padding = 50,
    btn0 = document.querySelector('#draw0'),
    btn1 = document.querySelector('#clear0'),
    input_r = document.querySelector('#r0'),
    input_dx = document.querySelector('#dx'),
    input_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    input_y1 = document.querySelector('#y1'),
    input_y2 = document.querySelector('#y2'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    p = (x) => pre0.textContent += x + '\n',
    range = (start, end, step=1) => {
        let res = [];
        for (let i = start; i < end; i += step) {
            res.push(i);
        }
        return res;
    };

let draw = () => {
    pre0.textContent = '';

    let r = parseFloat(input_r.value),
        dx = parseFloat(input_dx.value),
        x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [];
    
    for (let x = x1; x <= x2; x += dx) {
        let y = Math.sinh(x);
        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let t1 = points.length;

    for (let x = x1; x <= x2; x += dx) {
        let y = Math.cosh(x);
        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let t2 = points.length;

    for (let x = x1; x <= x2; x += dx) {
        let y = Math.tanh(x);
        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
   
    let xscale = d3.scaleLinear()
        .domain([x1, x2])
        .range([padding, width - padding]);

    let yscale = d3.scaleLinear()
        .domain([y1, y2])
        .range([height - padding, padding]);

    let xaxis = d3.axisBottom().scale(xscale);
    let yaxis = d3.axisLeft().scale(yscale);
    div0.innerHTML = '';
    let svg = d3.select('#graph0')
        .append('svg')
        .attr('width', width)
        .attr('height', height);

    svg.selectAll('circle')
        .data(points)
        .enter()
        .append('circle')
        .attr('cx', (d) => xscale(d[0]))
        .attr('cy', (d) => yscale(d[1]))
        .attr('r', r)
        .attr('fill', (d, i) =>
              i < t1 ? 'red':
              i < t2 ? 'green' : 'blue');

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]])
        .enter()
        .append('line')
        .attr('x1', (d) => xscale(d[0]))
        .attr('y1', (d) => yscale(d[1]))
        .attr('x2', (d) => xscale(d[2]))
        .attr('y2', (d) => yscale(d[3]))
        .attr('stroke', 'black');
    
    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);

    p('a.');
    range(0, 10).forEach((n, i) => {
        let x = (i % 2 === 0 ? 1 : -1) * Math.random() * 10;
        p(`cosh^2(${x}) - sinh^2(${x}) = ${Math.cosh(x) ** 2 - Math.sinh(x) ** 2}`);        
    });    
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

r = dx =
x1 = x2 =
y1 = y2 =