2017年9月2日土曜日


オイラーの贈物

学習環境

オイラーの贈物―人類の至宝eiπ=-1を学ぶ (吉田 武(著)、東海大学出版会)の第II部(関数の定義(Definitions of Functions))、第6章(指数関数・対数関数(Exponential & Logarithmic Functions))、6.2(指数関数)、6.2.1(指数関数のグラフ)、問題1.を取り組んでみる。


  1. d dx e − x 2 =−2x e − x 2 d 2 d x 2 e − x 2 =−2 e − x 2 −2x e − x 2 ( −2x ) =−2 e − x 2 +4 x 2 e − x 2 d 3 d x 3 e − x 2 =4x e − x 2 +8x e − x 2 +4 x 2 e − x 2 ( −2x ) =12x e − x 2 −8 x 3 e − x 2 H 0 = e x 2 e − x 2 =1 H 1 =− e x 2 ( −2x e − x 2 ) =2x H 2 = e x 2 ( −2 e − x 2 +4 x 2 e − x 2 ) =−2+4 x 2 H 3 =− e x 2 ( 12x e − x 2 −8 x 3 e − x 2 ) =−12x+8 x 3

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, exp, Derivative, plot

print('問題1')
x = symbols('x')

for n in range(4):
    Hn = (-1) ** n * exp(x ** 2) * Derivative(exp(-x ** 2), x, n)
    for g in [Hn, Hn.doit()]:
        pprint(g)
    try:
        p = plot(Hn.doit(), show=False, legend=True)
        p.save(f'sample1_{n}.svg')
    except Exception as err:
        print(type(err), err)
    print()

入出力結果(Terminal, IPython)

$ ./sample1.py
問題1
1
1
<class 'TypeError'> __init__() missing 1 required positional argument: 'var_start_end'

  ⎛ 2⎞   ⎛   2⎞
  ⎝x ⎠ d ⎜ -x ⎟
-ℯ    ⋅──⎝ℯ   ⎠
       dx      
2⋅x

 ⎛ 2⎞   2⎛   2⎞
 ⎝x ⎠  d ⎜ -x ⎟
ℯ    ⋅───⎝ℯ   ⎠
        2      
      dx       
   2    
4⋅x  - 2

  ⎛ 2⎞   3⎛   2⎞
  ⎝x ⎠  d ⎜ -x ⎟
-ℯ    ⋅───⎝ℯ   ⎠
         3      
       dx       
     ⎛     2    ⎞
-4⋅x⋅⎝- 2⋅x  + 3⎠

$

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="sample1.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 f0 = (x) => 1,
    f1 = (x) => 2 * x,
    f2 = (x) => 4 * x ** 2 - 2,
    f3 = (x) => -4 * x * (-2 * x ** 2 + 3);

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 = [],
        lines = [],
        fns = [[f0, 'red'],
               [f1, 'green'],
               [f2, 'blue'],
               [f3, 'orange']],
        fns1 = [],
        fns2 = [];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                if (Math.abs(y) < Infinity) {
                    points.push([x, y, color]);
                }
            }
        });                 

    fns2
        .forEach((o) => {
            let [f, color] = o;

            for (let x = x1; x <= x2; x += dx0) {
                let g = f(x);
                lines.push([x1, g(x1), x2, g(x2), color]);
            }
        });
    
    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('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
        .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', (d) => d[4] || 'black');

    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) => d[2] || 'green');

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

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

    [fns, fns1, fns2].forEach((fs) => p(fs.join('\n')));
    p(Math.sqrt(a0 + b0) - Math.sqrt(b0));
    p(2 * Math.sqrt(a0 + b0) / a0 -
      4 / (3 * a0 ** 2) * (Math.sqrt((a0 + b0) ** 3) - Math.sqrt(b0 ** 3)));
};

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



































						












時刻:


15:00




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