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LuxXx/noerw.ipynb

Created November 12, 2020 00:28
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noerw.ipynb
{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import sympy"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\[\n",
" f_n(x)=\\frac{n\\sin(\\frac{\\pi}{n})}{\\pi(1+x^n)}\n",
"\\]\n",
"\\[\n",
" F_n(t)=\\int_0^tf_n(x)\\,\\mathrm{d}x\n",
"\\]\n"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"%%latex\n",
"\\[\n",
" f_n(x)=\\frac{n\\sin(\\frac{\\pi}{n})}{\\pi(1+x^n)}\n",
"\\]\n",
"\\[\n",
" F_n(t)=\\int_0^tf_n(x)\\,\\mathrm{d}x\n",
"\\]"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\[\n",
" \\operatorname{E}(X^k)=\\int_0^\\infty x^k\\frac{n\\sin(\\frac{\\pi}{n})}{\\pi(1+x^n)}\\,\\mathrm{d}x=\\frac{\\sin(\\frac{\\pi}{n})}{\\sin(\\frac{\\pi}{n}(k+1))}\n",
"\\]\n"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"%%latex\n",
"\\[\n",
" \\operatorname{E}(X^k)=\\int_0^\\infty x^k\\frac{n\\sin(\\frac{\\pi}{n})}{\\pi(1+x^n)}\\,\\mathrm{d}x=\\frac{\\sin(\\frac{\\pi}{n})}{\\sin(\\frac{\\pi}{n}(k+1))}\n",
"\\]"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\begin{cases} \\frac{n \\sin{\\left(\\frac{\\pi}{n} \\right)}}{\\sin{\\left(\\frac{\\pi k}{n} + \\frac{\\pi}{n} \\right)} \\left|{n}\\right|} & \\text{for}\\: 2 - \\frac{k + 1}{n} > 1 \\wedge 1 - \\frac{k + 1}{n} < 1 \\\\\\int\\limits_{0}^{\\infty} \\frac{n x^{k} \\sin{\\left(\\frac{\\pi}{n} \\right)}}{\\pi \\left(x^{n} + 1\\right)}\\, dx & \\text{otherwise} \\end{cases}$"
],
"text/plain": [
"Piecewise((n*sin(pi/n)/(sin(pi*k/n + pi/n)*Abs(n)), (2 - (k + 1)/n > 1) & (1 - (k + 1)/n < 1)), (Integral(n*x**k*sin(pi/n)/(pi*(x**n + 1)), (x, 0, oo)), True))"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x = sympy.symbols('x')\n",
"n = sympy.symbols('n', integer=True)\n",
"k = sympy.symbols('k', integer=True)\n",
"f = 1 / (1+x**n)\n",
"f\n",
"sympy.integrate(sympy.sin(sympy.pi / n) * n / (sympy.pi) * x**k * f, (x, 0, sympy.oo))"
]
},
{
"cell_type": "code",
"execution_count": 55,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle 1$"
],
"text/plain": [
"1"
]
},
"execution_count": 55,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(sympy.sin(sympy.pi/n)/sympy.sin(sympy.pi/n * (k+1))).subs(n, 3).subs(k, 1)"
]
},
{
"cell_type": "code",
"execution_count": 65,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}0 & \\tilde{\\infty} & 1 & \\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{\\frac{5}{8} - \\frac{\\sqrt{5}}{8}}}{\\sqrt{\\frac{\\sqrt{5}}{8} + \\frac{5}{8}}} & \\frac{\\sqrt{3}}{3} & \\frac{\\sin{\\left(\\frac{\\pi}{7} \\right)}}{\\sin{\\left(\\frac{2 \\pi}{7} \\right)}} & \\sqrt{2} \\sqrt{\\frac{1}{2} - \\frac{\\sqrt{2}}{4}} & \\frac{\\sin{\\left(\\frac{\\pi}{9} \\right)}}{\\sin{\\left(\\frac{2 \\pi}{9} \\right)}} & \\frac{- \\frac{1}{4} + \\frac{\\sqrt{5}}{4}}{\\sqrt{\\frac{5}{8} - \\frac{\\sqrt{5}}{8}}}\\\\0 & 0 & \\tilde{\\infty} & 1 & \\frac{\\sqrt{\\frac{5}{8} - \\frac{\\sqrt{5}}{8}}}{\\sqrt{\\frac{\\sqrt{5}}{8} + \\frac{5}{8}}} & \\frac{1}{2} & \\frac{\\sin{\\left(\\frac{\\pi}{7} \\right)}}{\\sin{\\left(\\frac{3 \\pi}{7} \\right)}} & \\frac{\\sqrt{\\frac{1}{2} - \\frac{\\sqrt{2}}{4}}}{\\sqrt{\\frac{\\sqrt{2}}{4} + \\frac{1}{2}}} & \\frac{2 \\sqrt{3} \\sin{\\left(\\frac{\\pi}{9} \\right)}}{3} & \\frac{- \\frac{1}{4} + \\frac{\\sqrt{5}}{4}}{\\frac{1}{4} + \\frac{\\sqrt{5}}{4}}\\\\0 & 0 & 0 & \\tilde{\\infty} & 1 & \\frac{\\sqrt{3}}{3} & \\frac{\\sin{\\left(\\frac{\\pi}{7} \\right)}}{\\sin{\\left(\\frac{3 \\pi}{7} \\right)}} & \\sqrt{\\frac{1}{2} - \\frac{\\sqrt{2}}{4}} & \\frac{\\sin{\\left(\\frac{\\pi}{9} \\right)}}{\\sin{\\left(\\frac{4 \\pi}{9} \\right)}} & \\frac{- \\frac{1}{4} + \\frac{\\sqrt{5}}{4}}{\\sqrt{\\frac{\\sqrt{5}}{8} + \\frac{5}{8}}}\\\\0 & 0 & 0 & 0 & \\tilde{\\infty} & 1 & \\frac{\\sin{\\left(\\frac{\\pi}{7} \\right)}}{\\sin{\\left(\\frac{2 \\pi}{7} \\right)}} & \\frac{\\sqrt{\\frac{1}{2} - \\frac{\\sqrt{2}}{4}}}{\\sqrt{\\frac{\\sqrt{2}}{4} + \\frac{1}{2}}} & \\frac{\\sin{\\left(\\frac{\\pi}{9} \\right)}}{\\sin{\\left(\\frac{4 \\pi}{9} \\right)}} & - \\frac{1}{4} + \\frac{\\sqrt{5}}{4}\\\\0 & 0 & 0 & 0 & 0 & \\tilde{\\infty} & 1 & \\sqrt{2} \\sqrt{\\frac{1}{2} - \\frac{\\sqrt{2}}{4}} & \\frac{2 \\sqrt{3} \\sin{\\left(\\frac{\\pi}{9} \\right)}}{3} & \\frac{- \\frac{1}{4} + \\frac{\\sqrt{5}}{4}}{\\sqrt{\\frac{\\sqrt{5}}{8} + \\frac{5}{8}}}\\\\0 & 0 & 0 & 0 & 0 & 0 & \\tilde{\\infty} & 1 & \\frac{\\sin{\\left(\\frac{\\pi}{9} \\right)}}{\\sin{\\left(\\frac{2 \\pi}{9} \\right)}} & \\frac{- \\frac{1}{4} + \\frac{\\sqrt{5}}{4}}{\\frac{1}{4} + \\frac{\\sqrt{5}}{4}}\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tilde{\\infty} & 1 & \\frac{- \\frac{1}{4} + \\frac{\\sqrt{5}}{4}}{\\sqrt{\\frac{5}{8} - \\frac{\\sqrt{5}}{8}}}\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tilde{\\infty} & 1\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tilde{\\infty}\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{matrix}\\right]$"
],
"text/plain": [
"Matrix([\n",
"[0, zoo, 1, sqrt(2)/2, sqrt(5/8 - sqrt(5)/8)/sqrt(sqrt(5)/8 + 5/8), sqrt(3)/3, sin(pi/7)/sin(2*pi/7), sqrt(2)*sqrt(1/2 - sqrt(2)/4), sin(pi/9)/sin(2*pi/9), (-1/4 + sqrt(5)/4)/sqrt(5/8 - sqrt(5)/8)],\n",
"[0, 0, zoo, 1, sqrt(5/8 - sqrt(5)/8)/sqrt(sqrt(5)/8 + 5/8), 1/2, sin(pi/7)/sin(3*pi/7), sqrt(1/2 - sqrt(2)/4)/sqrt(sqrt(2)/4 + 1/2), 2*sqrt(3)*sin(pi/9)/3, (-1/4 + sqrt(5)/4)/(1/4 + sqrt(5)/4)],\n",
"[0, 0, 0, zoo, 1, sqrt(3)/3, sin(pi/7)/sin(3*pi/7), sqrt(1/2 - sqrt(2)/4), sin(pi/9)/sin(4*pi/9), (-1/4 + sqrt(5)/4)/sqrt(sqrt(5)/8 + 5/8)],\n",
"[0, 0, 0, 0, zoo, 1, sin(pi/7)/sin(2*pi/7), sqrt(1/2 - sqrt(2)/4)/sqrt(sqrt(2)/4 + 1/2), sin(pi/9)/sin(4*pi/9), -1/4 + sqrt(5)/4],\n",
"[0, 0, 0, 0, 0, zoo, 1, sqrt(2)*sqrt(1/2 - sqrt(2)/4), 2*sqrt(3)*sin(pi/9)/3, (-1/4 + sqrt(5)/4)/sqrt(sqrt(5)/8 + 5/8)],\n",
"[0, 0, 0, 0, 0, 0, zoo, 1, sin(pi/9)/sin(2*pi/9), (-1/4 + sqrt(5)/4)/(1/4 + sqrt(5)/4)],\n",
"[0, 0, 0, 0, 0, 0, 0, zoo, 1, (-1/4 + sqrt(5)/4)/sqrt(5/8 - sqrt(5)/8)],\n",
"[0, 0, 0, 0, 0, 0, 0, 0, zoo, 1],\n",
"[0, 0, 0, 0, 0, 0, 0, 0, 0, zoo],\n",
"[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])"
]
},
"execution_count": 65,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.Matrix(10, 10, lambda i,j: (sympy.sin(sympy.pi/n)/sympy.sin(sympy.pi/n * (k+1))).subs(n, j+1).subs(k, i+1) if j > i else 0)"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.8.2"
}
},
"nbformat": 4,
"nbformat_minor": 4
}
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