-
Notifications
You must be signed in to change notification settings - Fork 0
/
the-joy-of-experimental-mathematics.html
275 lines (263 loc) · 12.1 KB
/
the-joy-of-experimental-mathematics.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
<!DOCTYPE html>
<html lang="en">
<head>
<title>The joy of experimental mathematics - You don't need to prove this</title>
<link href="https://newptcai.github.io/feeds/all.atom.xml" type="application/atom+xml" rel="alternate" title="You don't need to prove this Full Atom Feed" />
<!-- CSS -->
<link rel="stylesheet" type="text/css" href="https://newptcai.github.io/theme/css/w3.css">
<link rel="stylesheet" type="text/css" href="https://newptcai.github.io/theme/css/style.css">
<link rel="stylesheet" type="text/css" href="https://newptcai.github.io/theme/css/jqcloud.css">
<link rel="stylesheet" type="text/css" href="https://newptcai.github.io/theme/css/all.min.css">
<link rel="stylesheet" type="text/css" href="https://newptcai.github.io/theme/css/shariff.min.css">
<link rel="stylesheet" type="text/css" href="https://newptcai.github.io/theme/css/pygments-highlight-github.css">
<!-- JavaScript -->
<script src="https://newptcai.github.io/theme/js/jquery-3.5.1.min.js"></script>
<script src="https://newptcai.github.io/theme/js/jqcloud.min.js"></script>
<!-- Meta -->
<meta charset="utf-8" />
<meta http-equiv="X-UA-Compatible" content="IE=edge" />
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
<meta name="HandheldFriendly" content="True" />
<meta name="author" content="Xing Shi Cai" />
<meta name="description" content="I came across an interesting identity on math.stackexchange.com." />
<meta name="keywords" content="analysis, Mathematica, CAS, experimental-math">
<!-- Facebook OpenGraph -->
<meta property="og:site_name" content="You don't need to prove this">
<meta property="og:title" content="The joy of experimental mathematics - You don't need to prove this" />
<meta property="og:description" content="I came across an interesting identity on math.stackexchange.com." />
<meta property="og:image" content="https://newptcai.github.io">
<meta property="og:type" content="article" />
<meta property="og:url" content="https://newptcai.github.io/the-joy-of-experimental-mathematics.html" />
<meta property="og:locale" content="de_DE" />
<meta property="og:locale:alternate" content="en_US" />
<!-- Twitter -->
<meta name="twitter:card" content="summary_large_image">
<meta name="twitter:title" content="The joy of experimental mathematics - You don't need to prove this">
<meta name="twitter:description" content="I came across an interesting identity on math.stackexchange.com.">
<meta name="twitter:image" content="https://newptcai.github.io">
</head>
<body>
<div class="w3-row w3-card w3-white">
<header id=banner>
<!-- AUTHOR INITIALS-->
<a href="https://newptcai.github.io" id=logo title="Home">XS</a>
<nav id="menu">
<ul>
<li><a href="https://newptcai.github.io/pages/research.html">Research</a></li>
<li><a href="https://newptcai.github.io/pages/teaching.html">Teaching</a></li>
<li class="active"><a href="https://newptcai.github.io/category/math.html">math</a></li>
<li ><a href="https://newptcai.github.io/category/mumble.html">mumble</a></li>
<li ><a href="https://newptcai.github.io/category/photo.html">photo</a></li>
</ul>
</nav>
</header>
</div>
<br>
<article>
<header class="w3-container col-main">
<h1>The joy of experimental mathematics</h1>
<div class="post-info">
<div class="w3-opacity w3-margin-right w3-margin-bottom" style="flex-grow: 1;">
<span> Posted on Sat 12 January 2019 in <a href="https://newptcai.github.io/category/math.html" style="font-style: italic">math</a>
</span>
</div>
<div id="article-tags">
<span class="w3-tag w3-light-grey w3-text-red w3-hover-red">
<a href="https://newptcai.github.io/tag/analysis.html" title=" All posts about Analysis
">#analysis</a>
</span>
<span class="w3-tag w3-light-grey w3-text-red w3-hover-red">
<a href="https://newptcai.github.io/tag/mathematica.html" title=" All posts about Mathematica
">#Mathematica</a>
</span>
<span class="w3-tag w3-light-grey w3-text-red w3-hover-red">
<a href="https://newptcai.github.io/tag/cas.html" title=" All posts about Cas
">#CAS</a>
</span>
<span class="w3-tag w3-light-grey w3-text-red w3-hover-red">
<a href="https://newptcai.github.io/tag/experimental-math.html" title=" All posts about Experimental-Math
">#experimental-math</a>
</span>
</div>
</div>
</header>
<br>
<div class="col-main w3-container">
<main id="article-content">
<p>I came across <a href="https://math.stackexchange.com/q/3070440/1618">an interesting identity</a> on math.stackexchange.com.
</p>
<div class="math">$$
\int_{0}^{1}((1-x^r)^{1/r}-x)^{2 n}\,\mathrm dx=\frac{1}{2 n+1}
$$</div>
<p>
for all <span class="math">\(n \in \mathbb N\)</span> and <span class="math">\(r > 0\)</span>.
Someone has found this <a href="https://math.stackexchange.com/a/3070493/1618">very smart proof</a> for it.</p>
<p>I will give a much less smart proof, but perhaps equally joyful one. When I see something like this,
I always try it with a few fix numbers with Mathematica. Let
</p>
<div class="math">$$
f(n)=\int \left(\left(1-x^r\right)^{1/r}-x\right)^{2 n} \, \mathrm dx
$$</div>
<p>
then,
</p>
<div class="math">$$
f(1)
=
x \, _2F_1\left(-\frac{2}{r},\frac{1}{r};1+\frac{1}{r};x^r\right)-x^2 \, _2F_1\left(-\frac{1}{r},\frac{2}{r};1+\frac{2}{r};x^r\right)+\frac{x^3}{3},
$$</div>
<div class="math">\begin{align}
f(2)
=
& x \, _2F_1\left(-\frac{4}{r},\frac{1}{r};1+\frac{1}{r};x^r\right)-x^4 \, _2F_1\left(-\frac{1}{r},\frac{4}{r};1+\frac{4}{r};x^r\right) \\
& +2 x^3 \, _2F_1\left(-\frac{2}{r},\frac{3}{r};1+\frac{3}{r};x^r\right)-2 x^2 \, _2F_1\left(-\frac{3}{r},\frac{2}{r};1+\frac{2}{r};x^r\right)+\frac{x^5}{5}
,
\end{align}</div>
<p>
and
</p>
<div class="math">\begin{align}
f(3)
=
&
x \, _2F_1\left(-\frac{6}{r},\frac{1}{r};1+\frac{1}{r};x^r\right)-x^6 \, _2F_1\left(-\frac{1}{r},\frac{6}{r};1+\frac{6}{r};x^r\right)
\\
&
+3 x^5 \, _2F_1\left(-\frac{2}{r},\frac{5}{r};1+\frac{5}{r};x^r\right)-5 x^4 \, _2F_1\left(-\frac{3}{r},\frac{4}{r};1+\frac{4}{r};x^r\right)
\\
&
+5 x^3 \, _2F_1\left(-\frac{4}{r},\frac{3}{r};1+\frac{3}{r};x^r\right)-3 x^2 \, _2F_1\left(-\frac{5}{r},\frac{2}{r};1+\frac{2}{r};x^r\right)+\frac{x^7}{7}
\end{align}</div>
<p>
where <span class="math">\(_2F_1\)</span> is the hypergeometric function.</p>
<p>Do you see the patterns? Looks like a binomial expansion, right? It turns out, if we do a binomail
expansion
</p>
<div class="math">$$
\left(\left(1-x^r\right)^{1/r}-x\right)^{2 n}
=
\underset{j=1}{\overset{2 n+1}{\sum }}
\left(
\begin{array}{c}
2 n \\
j-1 \\
\end{array}
\right)
(-x)^{j-1}
\left(\left(1-x^r\right)^{1/r}\right)^{-j+2 n+1}
$$</div>
<p>
Then it is easy to verify with Mathematica that
</p>
<div class="math">$$
\left(
\begin{array}{c}
2 n \\
j-1 \\
\end{array}
\right)
(-x)^{j-1}
\left(\left(1-x^r\right)^{1/r}\right)^{-j+2 n+1}
=
\frac{\mathrm d}{\mathrm d x}\left(
\frac{1}{2 n+1}
(-1)^{j+1} x^j \binom{2 n+1}{j} \, _2F_1\left(\frac{j}{r},-\frac{-j+2 n+1}{r};\frac{j}{r}+1;x^r\right)
\right)
$$</div>
<p>
Therefore, we have
</p>
<div class="math">$$
\int((1-x^r)^{1/r}-x)^{2 n} \mathrm dx
=
\sum _{j=1}^{2 n+1} \frac{1}{2 n+1} (-1)^{j+1} x^j \binom{2 n+1}{j} \, _2F_1\left(\frac{j}{r},-\frac{-j+2 n+1}{r};\frac{j}{r}+1;x^r\right).
$$</div>
<p>
When <span class="math">\(j=2n+1\)</span>, the summand in the right hand equals <span class="math">\(\frac{x^{2 n+1}}{2 n+1}\)</span>. This is the term
which gives us <span class="math">\(\frac 1 {2n+1}\)</span>. Everything else vanishes in the original definite integral.</p>
<p>The moral of this story is perhaps with the help of computers, it is much easier to find patterns
and often seeing the pattern is enough to give the proof. Moreover, the moment when you see the
pattern is quite joyful, as much as the moment that you come up with a proof.</p>
<script type="text/javascript">if (!document.getElementById('mathjaxscript_pelican_#%@#$@#')) {
var align = "center",
indent = "0em",
linebreak = "false";
if (false) {
align = (screen.width < 768) ? "left" : align;
indent = (screen.width < 768) ? "0em" : indent;
linebreak = (screen.width < 768) ? 'true' : linebreak;
}
var mathjaxscript = document.createElement('script');
mathjaxscript.id = 'mathjaxscript_pelican_#%@#$@#';
mathjaxscript.type = 'text/javascript';
mathjaxscript.src = 'https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.3/latest.js?config=TeX-AMS-MML_HTMLorMML';
var configscript = document.createElement('script');
configscript.type = 'text/x-mathjax-config';
configscript[(window.opera ? "innerHTML" : "text")] =
"MathJax.Hub.Config({" +
" config: ['MMLorHTML.js']," +
" TeX: { extensions: ['AMSmath.js','AMSsymbols.js','noErrors.js','noUndefined.js'], equationNumbers: { autoNumber: 'none' } }," +
" jax: ['input/TeX','input/MathML','output/HTML-CSS']," +
" extensions: ['tex2jax.js','mml2jax.js','MathMenu.js','MathZoom.js']," +
" displayAlign: '"+ align +"'," +
" displayIndent: '"+ indent +"'," +
" showMathMenu: true," +
" messageStyle: 'normal'," +
" tex2jax: { " +
" inlineMath: [ ['\\\\(','\\\\)'] ], " +
" displayMath: [ ['$$','$$'] ]," +
" processEscapes: true," +
" preview: 'TeX'," +
" }, " +
" 'HTML-CSS': { " +
" availableFonts: ['STIX', 'TeX']," +
" preferredFont: 'STIX'," +
" styles: { '.MathJax_Display, .MathJax .mo, .MathJax .mi, .MathJax .mn': {color: 'inherit ! important'} }," +
" linebreaks: { automatic: "+ linebreak +", width: '90% container' }," +
" }, " +
"}); " +
"if ('default' !== 'default') {" +
"MathJax.Hub.Register.StartupHook('HTML-CSS Jax Ready',function () {" +
"var VARIANT = MathJax.OutputJax['HTML-CSS'].FONTDATA.VARIANT;" +
"VARIANT['normal'].fonts.unshift('MathJax_default');" +
"VARIANT['bold'].fonts.unshift('MathJax_default-bold');" +
"VARIANT['italic'].fonts.unshift('MathJax_default-italic');" +
"VARIANT['-tex-mathit'].fonts.unshift('MathJax_default-italic');" +
"});" +
"MathJax.Hub.Register.StartupHook('SVG Jax Ready',function () {" +
"var VARIANT = MathJax.OutputJax.SVG.FONTDATA.VARIANT;" +
"VARIANT['normal'].fonts.unshift('MathJax_default');" +
"VARIANT['bold'].fonts.unshift('MathJax_default-bold');" +
"VARIANT['italic'].fonts.unshift('MathJax_default-italic');" +
"VARIANT['-tex-mathit'].fonts.unshift('MathJax_default-italic');" +
"});" +
"}";
(document.body || document.getElementsByTagName('head')[0]).appendChild(configscript);
(document.body || document.getElementsByTagName('head')[0]).appendChild(mathjaxscript);
}
</script>
</main>
<br>
<footer>
<div class="adjust-width">
<div id="author-block" class="w3-light-grey w3-border">
<img style="width: 35px; height: 56px; margin-left:50px;" src="https://newptcai.github.io/theme/images/bookmark-red.png" alt="bookmark"></img>
<div id="author-info">
<a href="https://newptcai.github.io/authors.html#xing-shi-cai"><img
style="width: 60px; height: 60px;" src="https://newptcai.github.io/authors/xing-shi-cai.png" onerror="this.src='https://newptcai.github.io/theme/images/avatar.png'"></img>
</a>
<div style="margin-left: 20px; margin-top: 15px;">
<a href="https://newptcai.github.io/authors.html#xing-shi-cai"><span id="author-name" class="w3-hover-text-dark-grey">Xing Shi Cai</span></a>
<p id="author-story" style="max-width: 500px;"></p>
</div>
</div>
</div>
</div>
<br>
</footer>
</div>
</article>
<br>
<script src="https://newptcai.github.io/theme/js/shariff.min.js"></script>
</body>
</html>