2017年7月10日月曜日

学習環境

解析入門〈2〉(松坂 和夫(著)、岩波書店)の第8章(積分の計算)、8.1(不定積分の計算)、問題4.を取り組んでみる。


    1. ( m+1 )I( m,n ) =( m+1 ) sin m x cos n xdx = ( m+1 ) sin m x cos n xdx = ( m+1 ) sin m xcosx cos n1 xdx = d dx ( sin m+1 x ) cos n1 xdx = sin m+1 x cos n1 x sin m+1 x( n1 ) cos n2 x( sinx )dx = sin m+1 x cos n1 +( n1 ) sin m+2 x cos n2 xdx = sin m+1 x cos n1 +( n1 ) sin m x cos n2 x sin 2 xdx = sin m+1 x cos n1 +( n1 ) sin m x cos n2 x( 1 cos 2 x )xdx = sin m+1 x cos n1 +( n1 )( sin m x cos n2 xdx sin m x cos n xdx ) = sin m+1 x cos n1 +( n1 )( I( m,n2 )I( m,n ) ) ( m+1 )I( m,n )= sin m+1 x cos n1 +( n1 )( I( m,n2 )I( m,n ) ) ( m+n )I( m,n )= sin m+1 x cos n1 +( n1 )I( m,n2 ) ( n+1 )I( m,n ) = ( n+1 ) sin m x cos n xdx = sin m1 x d dx ( cos n+1 x )dx = sin m1 cos n+1 x+ ( m1 ) sin m2 x cos n+2 xdx = sin m1 x cos n+1 x+( m1 ) sin m2 x cos n x cos 2 xdx = sin m1 x cos n+1 x+( m1 ) sin m2 x cos n x( 1 sin 2 x )dx = sin m1 x cos n+1 x+( m1 )( I( m2,n )I( m,n ) ) ( n+1 )I( m,n )= sin m1 x cos n+1 x+( m1 )( I( m2,n )I( m,n ) ) ( m+n )I( m,n )= sin m1 x cos n+1 x+( m1 )I( m2,n )

    2. m+n0 I( m,n )= sin m+1 x cos n1 m+n + n1 m+n I( m,n2 ) I( m,n )= sin m1 x cos n+1 x m+n + m1 m+n I( m2,n )

    3. ( m+n+2 )I( m,n+2 )= sin m+1 x cos n+1 x+( n+1 )I( m,n ) ( n+1 )I( m,n )= sin m+1 x cos n+1 x+( m+n+2 )I( m,n+2 ) n1 I( m,n )= sin m+1 x cos n+1 x n+1 + m+n+2 n+1 I( m,n+2 ) I( m+2,n )= sin m+1 x cos n+1 x m+n+2 + m+1 m+n+2 I( m,n ) ( m+1 )I( m,n )= sin m+1 x cos n+1 x+( m+n+2 )I( m+2,n ) m1 I( m,n )= sin m+1 x cos n+1 x m+1 + m+n+2 m+1 I( m+2,n )

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Integral, sin, cos, plot

print('4.')
m, n = symbols('m n', integer=True)
x = symbols('x')
I = Integral(sin(x) ** m + cos(x) ** n, x)
pprint(I)

m0 = 2
n0 = 3
eq = (I.subs({m: m0, n: n0}) - (
    sin(x) ** (m + 1) * cos(x) **
    (n - 1) / (m + n) + (n - 1) / (m + n) + I.subs(
        {m: m0, n: n0 - 2}))).subs({m: m0, n: n0})

pprint(eq)

入出力結果(Terminal, IPython)

$ ./sample4.py
4.
⌠                       
⎮ ⎛   m         n   ⎞   
⎮ ⎝sin (x) + cos (x)⎠ dx
⌡                       
     3       2      ⌠                         ⌠                           
  sin (x)⋅cos (x)   ⎮ ⎛   2            ⎞      ⎮ ⎛   2         3   ⎞      2
- ─────────────── - ⎮ ⎝sin (x) + cos(x)⎠ dx + ⎮ ⎝sin (x) + cos (x)⎠ dx - ─
         5          ⌡                         ⌡                          5
$         

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">
<br>
<label for="m0">m0 = </label>
<input id="m0" type="number" step="1" value="2">
<label for="n0">n0 = </label>
<input id="n0" type="number" step="1" value="3">
     
<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_m0 = document.querySelector('#m0'),
    input_n0 = document.querySelector('#n0'),    
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_m0, input_n0],
    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),
        m0 = parseFloat(input_m0.value),
        n0 = parseFloat(input_n0.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        lines = [],
        f = (x) => Math.sin(x) ** m0 * Math.cos(x) ** n0,
        fns = [[f, 'green']];

    fns
        .forEach((o) => {
            let [fn, color] = o;
            
            for (let x = x1; x <= x2; x += dx) {
                let y = fn(x);
                
                if (Math.abs(y) < Infinity) {
                    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('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.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.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

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

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








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