2017年5月25日木曜日

学習環境

解析入門〈1〉(松坂 和夫(著)、岩波書店)の第5章(各種の初等関数)、5.5(複素数の幾何学的表現)、問題1、2、3、4、5、6.を取り組んでみる。


  1. x α ¯ y α ¯ y=x i α ¯ y=x i α ¯

  2. αγ a'γ' = αβ α'β' αγ αβ = α'γ' α'β'

  3. αγ αβ = βα βγ αββγαγ+ γ 2 =αβ α 2 β 2 +αβ α 2 + β 2 + γ 2 =αβ+βγ+γα

  4. cos4θ+isin4θ= ( cosθ+isinθ ) 4 = cos 4 θ+4 cos 3 θisinθ6 cos 2 θsinθ4icosθ sin 3 θ+ sin 4 θ cos4θ= cos 4 θ6 cos 2 θsinθ+ sin 4 θ sin4θ=4 cos 3 θsinθ4cosθ sin 3 θ cos5θ+isin5θ= ( cosθ+isinθ ) 5 = cos 5 θ+5i cos 4 θsinθ10 cos 3 θ sin 2 θ10i cos 2 θ sin 3 θ+5cosθ sin 4 θi sin 5 θ cos5θ= cos 5 θ10 cos 3 θ sin 2 θ+5cosθ sin 4 θ sin5θ=5 cos 4 θsinθ10 cos 2 θ sin 3 θ sin 5 θ

  5. h=kn n hkn 0

  6. z=cosθ+isinθ 1 z n+1 1z = 1( cos( n+1 )θ+isin( n+1 )θ ) 1( cosθ+isinθ ) = ( 1cos( n+1 )θ )isin( n+1 )θ ( 1cosθ )isinθ = ( ( 1cos( n+1 )θ )isin( n+1 )θ )( ( 1cosθ )+isinθ ) ( 1cosθ ) 2 + sin 2 θ = ( 1cos( n+1 )θ )( 1cosθ )+sin( n+1 )θsinθ+i( ( 1cos( n+1 )θ )sinθsin( n+1 )θ( 1cosθ ) ) 12cosθ+ cos 2 θ+ sin 2 θ = 1+cos( n+1 )θcosθ+sin( n+1 )θsinθ+i( sinθcos( n+1 )θsinθsin( n+1 )θ+sin( n+1 )θcosθ ) 2( 1cosθ ) = 1+cos( ( n+1 )θ )+i( sinθsin( n+1 )θ+sin( ( n+1 )θ ) ) 2( 1cosθ ) 1+cos( ( n+1 )θ ) 2( 1cosθ ) sinθsin( n+1 )θ+sin( ( n+1 )θ ) 2( 1cosθ )

コード(Emacs)

Python 3

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

from sympy import symbols, pprint, conjugate, expand, cos, sin, pi, summation

print('1.')
a, b, c, d = symbols('a b c d', real=True)
alpha = a + b * 1J
pprint(alpha)
pprint(conjugate(alpha))
pprint(-conjugate(alpha))
pprint(expand(1j * conjugate(alpha)))
pprint(expand(-1j * conjugate(alpha)))

print('5')
h = symbols('h', integer=True)
n, i = symbols('n i', integer=True)
w = cos(2 * pi / n) + 1j * sin(2 * pi / n)

pprint(w)
s = summation(w ** (i * h), (i, 0, n - 1))
pprint(s)

print('6.')
theta = symbols('θ', real=True)
pprint(summation(cos(i * theta), (i, 0, n)))
pprint(summation(sin(i * theta), (i, 1, n)))

入出力結果(Terminal, IPython)

$ ./sample1.py
1.
a + 1.0⋅ⅈ⋅b
a - 1.0⋅ⅈ⋅b
-a + 1.0⋅ⅈ⋅b
1.0⋅ⅈ⋅a + 1.0⋅b
-1.0⋅ⅈ⋅a - 1.0⋅b
5
         ⎛2⋅π⎞      ⎛2⋅π⎞
1.0⋅ⅈ⋅sin⎜───⎟ + cos⎜───⎟
         ⎝ n ⎠      ⎝ n ⎠
⎧                                                                     h    
⎪                                          ⎛         ⎛2⋅π⎞      ⎛2⋅π⎞⎞     
⎪                 n                    for ⎜1.0⋅ⅈ⋅sin⎜───⎟ + cos⎜───⎟⎟  = 1
⎪                                          ⎝         ⎝ n ⎠      ⎝ n ⎠⎠     
⎪                                                                          
⎪                             h⋅n                                          
⎪  ⎛         ⎛2⋅π⎞      ⎛2⋅π⎞⎞                                             
⎨- ⎜1.0⋅ⅈ⋅sin⎜───⎟ + cos⎜───⎟⎟    + 1                                      
⎪  ⎝         ⎝ n ⎠      ⎝ n ⎠⎠                                             
⎪────────────────────────────────────               otherwise              
⎪                              h                                           
⎪   ⎛         ⎛2⋅π⎞      ⎛2⋅π⎞⎞                                            
⎪ - ⎜1.0⋅ⅈ⋅sin⎜───⎟ + cos⎜───⎟⎟  + 1                                       
⎪   ⎝         ⎝ n ⎠      ⎝ n ⎠⎠                                            
⎩                                                                          
6.
  n           
 ___          
 ╲            
  ╲   cos(i⋅θ)
  ╱           
 ╱            
 ‾‾‾          
i = 0         
  n           
 ___          
 ╲            
  ╲   sin(i⋅θ)
  ╱           
 ╱            
 ‾‾‾          
i = 1
$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="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">
<br>
α = <input id="a0" type="number" value="2"> + <input id="b0" type="number" value="3">i
<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.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_a = document.querySelector('#a0'),
    input_b = document.querySelector('#b0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_a, input_b],
    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),
        a = parseFloat(input_a.value),
        b = parseFloat(input_b.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [[a, b, 'red'],
                  [-a, b, 'green'],
                  [a, -b, 'blue'],
                  [b, a, 'orange'],
                  [-b, -a, 'brown']];
       
    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]);

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]])
        .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', 'black');
    
    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

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

};

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








α = + i

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