2017年7月13日木曜日

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7.を取り組んでみる。


  1. 2tan x 2 1+ tan 2 x 2 = 2 sin x 2 cos x 2 1+ sin 2 x 2 cos 2 x 2 =2sin x 2 cos x 2 =sinx 1 tan 2 x 2 1+ tan 2 x 2 = 1 sin 2 x 2 cos 2 x 2 1+ sin 2 x 2 cos 2 x 2 = cos 2 x 2 sin 2 x 2 =cosx dx dt = 1 dt dx = 1 1 cos 2 x 2 · 1 2 = 2 sin 2 x 2 + cos 2 x 2 cos 2 x 2 = 2 1+ tan 2 x 2 = 2 1+ t 2

    1. 1 1+ 1 t 2 1+ t 2 · 2 1+ t 2 dt = 2 1+ t 2 +1 t 2 ·dt = 1dt =t =tan x 2

    2. 1 1 t 2 1+ t 2 · 2 1+ t 2 dt = 2 1 t 2 dt = 2 ( 1+t )( 1t ) dt A 1+t + B 1t = ( BA )t+A+B ( 1+t )( 1t ) BA=0 A+B=2 A=B 2A=2 A=1 B=1 2 ( 1+t )( 1t ) dt = 1 1+t dt + 1 1t dt =log| 1+t |log| 1t | =log| 1+t 1t | =log| 1+tan x 2 1tan x 2 |

    3. 2t 1+ t 2 1+ 2t 1+ t 2 · 2 1+ t 2 dt = 2t 1+ t 2 +2t · 2 1+ t 2 dt =4 t ( t+1 ) 2 ( t 2 +1 ) dt A t+1 + B ( t+1 ) 2 + Ct+D t 2 +1 = A( t+1 )( t 2 +1 )+B( t 2 +1 )+( Ct+D ) ( t+1 ) 2 ( t+1 ) 2 ( t 2 +1 ) A( t+1 )( t 2 +1 )+B( t 2 +1 )+( Ct+D ) ( t+1 ) 2 =A( t 3 + t 2 +t+1 )+B t 2 +B+( Ct+D )( t 2 +2t+1 ) =( A+C ) t 3 +( A+B+2C+D ) t 2 +( A+C+2D )t+A+B+D A+C=0 A+B+2C+D=0 A+C+2D=1 A+B+D=0 C=A A+B2A+D=0 AA+2D=1 A+B+D=0 D=AB A+B2AAB=0 A=0 D=B 2D=1 D= 1 2 B= 1 2 C=0 4 t ( t+1 ) 2 ( t 2 +1 ) dt =4 ( 1 2 ( t+1 ) 2 + 1 2 t 2 +1 )dt =2 ( 1 ( t+1 ) 2 + 1 t 2 +1 )dt =2( ( t+1 ) 1 +arctant ) =2( 1 tan x 2 +1 +arctan( tan x 2 ) ) = 2 tan x 2 +1 +2 x 2 = 2 tan x 2 +1 +x

コード(Emacs)

Python 3

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

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

print('7.')
x = symbols('x')

fs = [1 / (1 + cos(x)),
      1 / cos(x),
      sin(x) / (1 + sin(x))]


for i, f in enumerate(fs, 1):
    print(f'({i})')
    I = Integral(f, x)
    pprint(I)
    I = I.doit()
    pprint(I)
    print('factor:')
    pprint(I.factor())
    print('expand:')
    pprint(I.expand())

入出力結果(Terminal, IPython)

$ ./sample7.py
7.
(1)
⌠              
⎮     1        
⎮ ────────── dx
⎮ cos(x) + 1   
⌡              
   ⎛x⎞
tan⎜─⎟
   ⎝2⎠
factor:
   ⎛x⎞
tan⎜─⎟
   ⎝2⎠
expand:
   ⎛x⎞
tan⎜─⎟
   ⎝2⎠
(2)
⌠          
⎮   1      
⎮ ────── dx
⎮ cos(x)   
⌡          
  log(sin(x) - 1)   log(sin(x) + 1)
- ─────────────── + ───────────────
         2                 2       
factor:
-log(sin(x) - 1) + log(sin(x) + 1)
──────────────────────────────────
                2                 
expand:
  log(sin(x) - 1)   log(sin(x) + 1)
- ─────────────── + ───────────────
         2                 2       
(3)
⌠              
⎮   sin(x)     
⎮ ────────── dx
⎮ sin(x) + 1   
⌡              
      ⎛x⎞                           
 x⋅tan⎜─⎟                           
      ⎝2⎠        x            2     
────────── + ────────── + ──────────
   ⎛x⎞          ⎛x⎞          ⎛x⎞    
tan⎜─⎟ + 1   tan⎜─⎟ + 1   tan⎜─⎟ + 1
   ⎝2⎠          ⎝2⎠          ⎝2⎠    
factor:
     ⎛x⎞        
x⋅tan⎜─⎟ + x + 2
     ⎝2⎠        
────────────────
      ⎛x⎞       
   tan⎜─⎟ + 1   
      ⎝2⎠       
expand:
      ⎛x⎞                           
 x⋅tan⎜─⎟                           
      ⎝2⎠        x            2     
────────── + ────────── + ──────────
   ⎛x⎞          ⎛x⎞          ⎛x⎞    
tan⎜─⎟ + 1   tan⎜─⎟ + 1   tan⎜─⎟ + 1
   ⎝2⎠          ⎝2⎠          ⎝2⎠
$   

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="-10">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-10">
<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="sample7.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 f1 = (x) => 1 / (1 + Math.cos(x)),
    f2 = (x) => 1 / Math.cos(x),
    f3 = (x) => Math.sin(x) / (1 + Math.sin(x));

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 = [[f1, 'red'],
               [f2, 'green'],
               [f3, 'blue']];

    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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