2017年7月9日日曜日

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2、3.を取り組んでみる。


  1. P( x ) Q( x ) = k=1 n A k x a k P( x )=Q( x ) k=1 n A k x a k = k=1 n A k ( x a 1 )( x a k1 )( x a k+1 )( x a n ) P( a k )= A k ( x a 1 )( x a k1 )( x a k+1 )( x a n ) = A k Q'( a k ) a k = A k Q'( a k ) Q'( a k ) a k = A k P( x ) Q( x ) = k=1 n a k x a k P( x ) Q( x ) dx = k=1 n a k x a k dx = k=1 n a k ( 1 x a k dx ) = k=1 n ( a k log| x a k | ) = k=1 n ( log | x a k | a k ) dx =log| k=1 n ( x a k ) a k |

    1. x 2 +a dx =x x 2 +a x 1 2 ( x 2 +a ) 1 2 2xdx =x x 2 +a x 2 x 2 +a dx =x x 2 +a x 2 +aa x 2 +a dx =x x 2 +a ( x 2 +a dx a x 2 +a dx ) 2 x 2 +a dx =x x 2 +a a x 2 +a dx x 2 +a dx = 1 2 ( x x 2 +a alog| x+ x 2 +a | )

    2. 1x 1 x 2 dx = 1 1 x 2 dx x 1 x 2 dx =arcsinx+ 1 x 2

    3. x= 1 t dx dt = 1 t 2 dx= 1 t 2 dt 1 1 t 1 t 2 +1 1 t 2 dt = 1 1+ t 2 dt =log| t+ t 2 +1 | =log| 1 x + 1 x 2 +1 | =log| 1+ 1+ x 2 x | =log| x 1+ 1+ x 2 |

    4. x+1 x1 =t dt= 1 2 ( ( x+1 x1 ) 1 2 x1( x+1 ) ( x1 ) 2 )dx = 1 2 · x1 x+1 2 ( x1 ) 2 dx = x1 x+1 1 ( x1 ) 2 dx = x1 x 2 1 ( x1 ) 2 dx = 1 ( x1 ) x 2 1 dx 1 ( x1 ) x 2 1 dx = dt =t = x+1 x1

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Integral, sqrt

print('3.')
x, a = symbols('x a')
fs = [sqrt(x ** 2 + a),
      sqrt((1 - x) / (1 + x)),
      1 / (x * sqrt(x ** 2 + 1)),
      1 / ((x - 1) * sqrt(x ** 2 - 1))]


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

入出力結果(Terminal, IPython)

$ ./sample2.py
3.
(1)
⌠               
⎮    ________   
⎮   ╱      2    
⎮ ╲╱  a + x   dx
⌡               
          ________              
         ╱      2               
        ╱      x            ⎛x ⎞
√a⋅x⋅  ╱   1 + ──    a⋅asinh⎜──⎟
     ╲╱        a            ⎝√a⎠
────────────────── + ───────────
        2                 2     

(2)
⌠                
⎮     ________   
⎮    ╱ -x + 1    
⎮   ╱  ──────  dx
⎮ ╲╱   x + 1     
⌡                
⌠                
⎮     ________   
⎮    ╱ -x + 1    
⎮   ╱  ──────  dx
⎮ ╲╱   x + 1     
⌡                

(3)
⌠                 
⎮       1         
⎮ ───────────── dx
⎮      ________   
⎮     ╱  2        
⎮ x⋅╲╱  x  + 1    
⌡                 
      ⎛1⎞
-asinh⎜─⎟
      ⎝x⎠

(4)
⌠                       
⎮          1            
⎮ ─────────────────── dx
⎮            ________   
⎮           ╱  2        
⎮ (x - 1)⋅╲╱  x  - 1    
⌡                       
⌠                               
⎮              1                
⎮ ─────────────────────────── dx
⎮   _________________           
⎮ ╲╱ (x - 1)⋅(x + 1) ⋅(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">
<br>
<label for="a0">a0 = </label>
<input id="a0" type="number" value="2">

     
<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="sample2.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_a0 = document.querySelector('#a0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
             input_a0],
    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 f2 = (x) => Math.sqrt((1 - x) / (1 + x)),
    f3 = (x) => 1 / (x * Math.sqrt(x ** 2 + 1)),
    f4 = (x) => 1 / ((x - 1) * Math.sqrt(x ** 2 - 1));

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),
        a0 = parseFloat(input_a0.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        lines = [],
        f1 = (x) => Math.sqrt(x ** 2 + a0),
        fns = [[f1, 'red'],
               [f2, 'green'],
               [f3, 'blue'],
               [f4, 'orange']];

    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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