数学 - Python - JavaScript - 解析学 - 積分 - 積分の計算 - いろいろな問題(部分積分法、広義積分、累乗関数、指数関数、積、帰納法)

学習環境

解析入門原書第3版 (S.ラング(著)、松坂和夫(翻訳)、片山孝次(翻訳)、岩波書店)の第3部(積分)、第11章(積分の計算)、補充問題(いろいろな問題)4.を取り組んでみる。

$\begin{array}{l} I_{0} = \int_{0}^{\infty} x^{0} e^{- x} dx \\ = \int_{0}^{\infty} e^{- x} dx \\ = {[- e^{- x}]}_{0}^{\infty} \\ = 1 \\ I_{1} = \int_{0}^{\infty} x^{1} e^{(- x)} dx \\ = {[- x e^{(- x)}]}_{0}^{\infty} + \int_{0}^{\infty} e^{(- x)} dx \\ = 1 \\ I_{2} = \int_{0}^{\infty} x^{2} e^{- x} dx \\ = {[- x^{2} e^{- x}]}_{0}^{\infty} + 2 \int_{0}^{\infty} x e^{- x} dx \\ = 2 \\ I_{3} = \int_{0}^{\infty} x^{3} e^{- x} dx \\ = {[- x^{3} e^{- x}]}_{0}^{\infty} + 3 \int_{0}^{\infty} x^{2} e^{- x} dx \\ = 3 \cdot 2 \\ = 6 \end{array}$

以上から

$I_{n} = n!$

と予想。

$\begin{array}{l} I_{n} = \int_{0}^{\infty} x^{n} e^{- x} dx \\ = {[- x^{n} e^{- x}]}_{0}^{\infty} + n \int_{0}^{\infty} x^{(n - 1)} e^{(- x)} dx \\ = n (n - 1)! \end{array}$

ゆえに、帰納法より、推測通り、

$\int_{0}^{\infty} x^{n} e^{- x} dx = n!$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Integral, plot, exp, oo

print('4.')

x = symbols('x')
n = symbols('n', integer=True, nonnegative=True)
f = x ** n * exp(-x)
I = Integral(f, (x, 0, oo))
d = {n: 3}
for t in [I, I.doit()]:
    pprint(t)
    print()
p = plot(f.subs(d), I.doit().subs(d), xlim=(0, 10),
         ylim=(0, 10), legend=True, show=False)
colors = ['red', 'green']
for i, c in enumerate(colors):
    p[i].line_color = c
p.save('sample4.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample4.py
4.
∞          
⌠          
⎮  n  -x   
⎮ x ⋅ℯ   dx
⌡          
0          

Γ(n + 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="0.001" value="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="0">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">

<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="sample4.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'),
    input_n0 = document.querySelector('#n0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    p = (x) => pre0.textContent += x + '\n';

let f = (x) => x ** 3 * Math.exp(-x),
    fns = [[f, 'red']];

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 = [[x1, 6, x2, 6, 'green']];

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

                points.push([x, y, 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].forEach((fs) => p(fs.join('\n')));
};

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

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

Kamimura's blog

2018年10月9日火曜日

数学 - Python - JavaScript - 解析学 - 積分 - 積分の計算 - いろいろな問題(部分積分法、広義積分、累乗関数、指数関数、積、帰納法)

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