{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Install"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "To load \"drakma\":\n",
      "  Load 1 ASDF system:\n",
      "    drakma\n",
      "\n",
      "; Loading \"drakma\"\n",
      "\n",
      "To load \"maxima-asdf\":\n",
      "  Load 1 ASDF system:\n",
      "    maxima-asdf\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(:MAXIMA-ASDF)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "; Loading \"maxima-asdf\"\n",
      "\n"
     ]
    }
   ],
   "source": [
    ":lisp (progn (ql:quickload :drakma)(ql:quickload :maxima-asdf))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "install_github(\"YasuakiHonda\",\"euler-maclaurin-sum\",\"master\")$\n",
    "asdf_load_source(\"euler-maclaurin-sum\")$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Variable ems"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{5}$}\\sum_{n=N}^{M}{f\\left(n\\right)}=\\sum_{k=1}^{K-1}{\\frac{\\left(-1\\right)^{k+1}\\,B_{k+1}\\,\\left(\\left.\\frac{d^{k}}{d\\,x^{k}}\\,f\\left(x\\right)\\right|_{x=M}-\\left.\\frac{d^{k}}{d\\,x^{k}}\\,f\\left(x\\right)\\right|_{x=N}\\right)}{\\left(k+1\\right)!}}+\\frac{\\left(-1\\right)^{K+1}\\,\\int_{N}^{M}{\\overline{B}_{K}\\left(x\\right)\\,\\left(\\frac{d^{K}}{d\\,x^{K}}\\,f\\left(x\\right)\\right)\\;dx}}{K!}+\\int_{N}^{M}{f\\left(x\\right)\\;dx}+\\frac{f\\left(N\\right)}{2}+\\frac{f\\left(M\\right)}{2}\\]"
      ],
      "text/plain": [
       "       M\n",
       "      ====\n",
       "      \\\n",
       "(%o5)  >    f(n) = \n",
       "      /\n",
       "      ====\n",
       "      n = N\n",
       "                                        !                  !\n",
       "                               k        !         k        !\n",
       "           k + 1              d         !        d         !\n",
       "      (- 1)      bern(k + 1) (--- (f(x))!      - --- (f(x))!     )\n",
       "K - 1                           k       !          k       !\n",
       "====                          dx        !        dx        !\n",
       "\\                                       !x = M             !x = N\n",
       " >    ------------------------------------------------------------\n",
       "/                               (k + 1)!\n",
       "====\n",
       "k = 1\n",
       "               M\n",
       "              /                            K\n",
       "        K + 1 [                           d\n",
       "   (- 1)      I  periodic_bernpoly(x, K) (--- (f(x))) dx\n",
       "              ]                             K               M\n",
       "              /                           dx               /\n",
       "               N                                           [            f(N)\n",
       " + ----------------------------------------------------- + I  f(x) dx + ----\n",
       "                            K!                             ]             2\n",
       "                                                           /\n",
       "                                                            N\n",
       "   f(M)\n",
       " + ----\n",
       "    2"
      ],
      "text/x-maxima": [
       "'sum(f(n),n,N,M) = 'sum(((-1)^(k+1)*bern(k+1)\n",
       "                                   *('at('diff(f(x),x,k),x = M)\n",
       "                                    -'at('diff(f(x),x,k),x = N)))\n",
       "                         /(k+1)!,k,1,K-1)\n",
       "                 +((-1)^(K+1)*'integrate(\n",
       "                              periodic_bernpoly(x,K)*'diff(f(x),x,K),x,N,M))\n",
       "                  /K!+'integrate(f(x),x,N,M)+f(N)/2+f(M)/2"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ems;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Specifying definition of f(n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Is N - 1 positive, negative or zero?\n",
      "pos;\n",
      "\n",
      "Is N - M + 1 positive, negative or zero?\n",
      "neg;\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{8}$}\\sum_{n=N}^{M}{\\log n}=\\sum_{k=1}^{K-1}{\\frac{\\left(-1\\right)^{k+1}\\,B_{k+1}\\,\\left(\\left.\\frac{d^{k}}{d\\,x^{k}}\\,\\log x\\right|_{x=M}-\\left.\\frac{d^{k}}{d\\,x^{k}}\\,\\log x\\right|_{x=N}\\right)}{\\left(k+1\\right)!}}+\\frac{\\left(-1\\right)^{K+1}\\,\\int_{N}^{M}{\\overline{B}_{K}\\left(x\\right)\\,\\left(\\frac{d^{K}}{d\\,x^{K}}\\,\\log x\\right)\\;dx}}{K!}-N\\,\\log N+\\frac{\\log N}{2}+N+M\\,\\log M+\\frac{\\log M}{2}-M\\]"
      ],
      "text/plain": [
       "       M\n",
       "      ====\n",
       "      \\\n",
       "(%o8)  >    log(n) = \n",
       "      /\n",
       "      ====\n",
       "      n = N\n",
       "                                          !                    !\n",
       "                               k          !         k          !\n",
       "           k + 1              d           !        d           !\n",
       "      (- 1)      bern(k + 1) (--- (log(x))!      - --- (log(x))!     )\n",
       "K - 1                           k         !          k         !\n",
       "====                          dx          !        dx          !\n",
       "\\                                         !x = M               !x = N\n",
       " >    ----------------------------------------------------------------\n",
       "/                                 (k + 1)!\n",
       "====\n",
       "k = 1\n",
       "               M\n",
       "              /                            K\n",
       "        K + 1 [                           d\n",
       "   (- 1)      I  periodic_bernpoly(x, K) (--- (log(x))) dx\n",
       "              ]                             K\n",
       "              /                           dx\n",
       "               N                                                        log(N)\n",
       " + ------------------------------------------------------- - N log(N) + ------\n",
       "                             K!                                           2\n",
       "                  log(M)\n",
       " + N + M log(M) + ------ - M\n",
       "                    2"
      ],
      "text/x-maxima": [
       "'sum(log(n),n,N,M) = 'sum(((-1)^(k+1)*bern(k+1)\n",
       "                                     *('at('diff(log(x),x,k),x = M)\n",
       "                                      -'at('diff(log(x),x,k),x = N)))\n",
       "                           /(k+1)!,k,1,K-1)\n",
       "                   +((-1)^(K+1)*'integrate(\n",
       "                                periodic_bernpoly(x,K)*'diff(log(x),x,K),x,N,\n",
       "                                M))\n",
       "                    /K!-N*log(N)+log(N)/2+N+M*log(M)+log(M)/2-M"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n"
     ]
    }
   ],
   "source": [
    "ems,f(x):=log(x),nouns;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Specifying values of free variables N,M,K"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Is M - 1 positive, negative or zero?\n",
      "pos;\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{27}$}\\sum_{n=1}^{M}{\\log n}=\\frac{\\int_{1}^{M}{\\frac{\\overline{B}_{5}\\left(x\\right)}{x^5}\\;dx}}{5}+M\\,\\log M+\\frac{\\log M}{2}-M+\\frac{\\frac{1}{M}-1}{12}-\\frac{\\frac{2}{M^3}-2}{720}+1\\]"
      ],
      "text/plain": [
       "                       M\n",
       "                      /\n",
       "                      [  periodic_bernpoly(x, 5)\n",
       "                      I  ----------------------- dx\n",
       "        M             ]             5\n",
       "       ====           /            x\n",
       "       \\               1                                         log(M)\n",
       "(%o27)  >    log(n) = ----------------------------- + M log(M) + ------ - M\n",
       "       /                            5                              2\n",
       "       ====\n",
       "       n = 1\n",
       "                                                                     2\n",
       "                                                             1       -- - 2\n",
       "                                                             - - 1    3\n",
       "                                                             M       M\n",
       "                                                           + ----- - ------ + 1\n",
       "                                                              12      720"
      ],
      "text/x-maxima": [
       "'sum(log(n),n,1,M) = ('integrate(periodic_bernpoly(x,5)/x^5,x,1,M))/5\n",
       "                   +M*log(M)+log(M)/2-M+(1/M-1)/12-(2/M^3-2)/720+1"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n"
     ]
    }
   ],
   "source": [
    "LOGSUM:ems,f(x):=log(x),N:1,K:5,nouns;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Give definition for periodic Bernulli polinomial"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{28}$}\\left[ \\overline{B}_{n}\\left(x\\right):={\\it bernpoly}\\left(x-\\left \\lfloor x \\right \\rfloor , n\\right) \\right] \\]"
      ],
      "text/plain": [
       "(%o28)      [periodic_bernpoly(x, n) := bernpoly(x - floor(x), n)]"
      ],
      "text/x-maxima": [
       "[periodic_bernpoly(x,n):=bernpoly(x-floor(x),n)]"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def_pbp();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{31}$}\\mathbf{done}\\]"
      ],
      "text/plain": [
       "(%o31)                               done"
      ],
      "text/x-maxima": [
       "done"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "undef_pbp();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Transform remainder term into sum of integration of Bernulli polinomial"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{60}$}\\int_{a}^{b}{\\frac{\\overline{B}_{n}\\left(x\\right)}{t\\left(x\\right)}\\;dx}\\]"
      ],
      "text/plain": [
       "                          b\n",
       "                         /\n",
       "                         [  periodic_bernpoly(x, n)\n",
       "(%o60)                   I  ----------------------- dx\n",
       "                         ]           t(x)\n",
       "                         /\n",
       "                          a"
      ],
      "text/x-maxima": [
       "'integrate(periodic_bernpoly(x,n)/t(x),x,a,b)"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(periodic_bernpoly(x,n)/t(x),x,a,b);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{59}$}\\sum_{k=a}^{b-1}{\\int_{k}^{k+1}{\\frac{{\\it bernpoly}\\left(x-k , n\\right)}{t\\left(x\\right)}\\;dx}}\\]"
      ],
      "text/plain": [
       "                      b - 1  k + 1\n",
       "                      ====  /\n",
       "                      \\     [      bernpoly(x - k, n)\n",
       "(%o59)                 >    I      ------------------ dx\n",
       "                      /     ]             t(x)\n",
       "                      ====  /\n",
       "                      k = a  k"
      ],
      "text/x-maxima": [
       "'sum('integrate(bernpoly(x-k,n)/t(x),x,k,k+1),k,a,b-1)"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "apply1(%,emsRemInt);"
   ]
  }
 ],
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