2017年7月26日水曜日

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)、8.2(定積分の計算)、問題1-7.を取り組んでみる。


    1. 部分積分法を繰り返す。

      e ax sinbxdx = 1 a e ax sinbx ( 1 a e ax bcosbx )dx = e ax sinbx a + b a e ax cosbxdx = e ax sinbx a + b a ( e ax a cosbx ( e ax a bsinbx )dx ) = e ax sinbx a + b a ( e ax a cosbx b a ( e ax sinbx )dx ) = e ax sinbx a b e ax cosbx a 2 b 2 a 2 e ax sinbxdx e ax sinbxdx = e ax sinbx a b e ax cosbx a 2 1+ b 2 a 2 = a e ax sinbxb e ax cosbx a 2 + b 2 0 + e ax sinbxdx = [ a e ax sinbxb e ax cosbx a 2 + b 2 ] 0 + = b a 2 + b 2

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Integral, exp, sin, S, plot

print('1.')
x = symbols('x')
a = symbols('a', positive=True)
b = symbols('b', nonzero=True)

fs = [(exp(- a * x) * sin(b * x), (0, S.Infinity))]

for i, (f, (x1, x2)) in enumerate(fs, 6):
    print(f'({i})')
    I = Integral(f, (x, x1, x2))
    pprint(I)
    I = I.doit()
    pprint(I)
    print('factor:')
    pprint(I.factor())
    print('expand:')
    pprint(I.expand())

f = fs[0][0]
p = plot(f.subs({a: 2, b: -3}), show=False, legend=True)
p.save('sample1_7.svg')

入出力結果(Terminal, IPython)

$ ./sample1_7.py
1.
(6)
∞                  
⌠                  
⎮  -a⋅x            
⎮ ℯ    ⋅sin(b⋅x) dx
⌡                  
0                  
⎧        1                │                 ⎛          2      ⎞│    
⎪    ──────────       for │periodic_argument⎝polar_lift (b), ∞⎠│ = 0
⎪      ⎛ 2    ⎞                                                     
⎪      ⎜a     ⎟                                                     
⎪    b⋅⎜── + 1⎟                                                     
⎪      ⎜ 2    ⎟                                                     
⎪      ⎝b     ⎠                                                     
⎨                                                                   
⎪∞                                                                  
⎪⌠                                                                  
⎪⎮  -a⋅x                                                            
⎪⎮ ℯ    ⋅sin(b⋅x) dx                    otherwise                   
⎪⌡                                                                  
⎪0                                                                  
⎩                                                                   
factor:
⎧        1                │                 ⎛          2      ⎞│    
⎪      ──────         for │periodic_argument⎝polar_lift (b), ∞⎠│ = 0
⎪       2                                                           
⎪      a                                                            
⎪      ── + b                                                       
⎪      b                                                            
⎨                                                                   
⎪∞                                                                  
⎪⌠                                                                  
⎪⎮  -a⋅x                                                            
⎪⎮ ℯ    ⋅sin(b⋅x) dx                    otherwise                   
⎪⌡                                                                  
⎩0                                                                  
expand:
⎧        1                │                 ⎛          2      ⎞│    
⎪      ──────         for │periodic_argument⎝polar_lift (b), ∞⎠│ = 0
⎪       2                                                           
⎪      a                                                            
⎪      ── + b                                                       
⎪      b                                                            
⎨                                                                   
⎪∞                                                                  
⎪⌠                                                                  
⎪⎮  -a⋅x                                                            
⎪⎮ ℯ    ⋅sin(b⋅x) dx                    otherwise                   
⎪⌡                                                                  
⎩0                                                                  
$

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">a = </label>
<input id="a0" type="number" min="0" value="1">
<label for="b0">b = </label>
<input id="b0" type="number" min="0" 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="sample1_7.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'),
    input_b0 = document.querySelector('#b0'),    
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_a0, input_b0],
    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 f = (x) => x * Math.asin(x) / Math.sqrt(1 - x ** 2);

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),
        b0 = parseFloat(input_b0.value);
        
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        y3 = b0 / (a0 ** 2 + b0 ** 2),
        lines = [[x1, y3, x2, y3, 'blue']],
        f = (x) => Math.exp(-a0 * x) * Math.sin(b0 * x),
        g = (x) => (-a0 * Math.exp(-a0 * x) * Math.sin(b0 * x) - b0 * Math.exp(-a0 * x) * Math.cos(b0 * x)) / (a0 ** 2 + b0 ** 2),
        h = (x) => g(x) - g(0),
        fns = [[f, 'green'],
               [h, '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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