1. Introduction
Summation is widely used in physics, especially in statistical physics. A traditional but powerful tool to work out a summation is the Euler–Maclaurin formula [1, 2], which has lots of generalizations [3–5]. If a real function f is Cn on an interval $[a^{\prime} ,b]$ for some positive integer n and integers $a^{\prime} \lt b$, then the Euler–Maclaurin formula reads2 ) and throughout stand for the floor function. The remainder term is1 ), one obtains an approximation of the summation on the left hand side. It is widely used for its high efficiency in most cases [7–9]. However, as most asymptotic expansions, for given $a^{\prime} ,b$ and f, the remainder term usually does not converge to 0 as n → + ∞ . In our recent work [10], a modified Euler–Maclaurin formula (MEMF) was introduced to keep part of the remainder term in order to achieve higher accuracy. However, the MEMF was expressed rather technically than rigorously there. In this paper, we first review the MEMF in section 2 in a more rigorous way including an estimation of the upper limit of the remainder term, and then generalize the result to Dimension 2 in section 3 . Some applications of the formulae in statistical physics are given in section 4 .
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \sum _{i=a^{\prime} }^{b}f(i) & = & {\displaystyle \int }_{a^{\prime} }^{b}f(x){\rm{d}}x+\displaystyle \frac{f(a^{\prime} )+f(b)}{2}\\ & & -\displaystyle \sum _{r=1}^{\rlap{-}n}\displaystyle \frac{{B}_{2r}}{(2r)!}{M}_{2r-1}(a^{\prime} ,b)+{R}_{n}.\end{array}\end{eqnarray}$
Here B2r are the 2rth Bernoulli number and the symbols $n-$ and M are defined as $\begin{eqnarray}\begin{array}{rcl}\rlap{-}n & = & \left[\displaystyle \frac{n}{2}\right],\\ {M}_{\nu }(a^{\prime} ,b) & = & {f}^{(\nu )}(a^{\prime} )-{f}^{(\nu )}(b),\end{array}\end{eqnarray}$
respectively, where the brakets in equation ( $\begin{eqnarray}{R}_{n}=\displaystyle \frac{{\left(-1\right)}^{n+1}}{n!}{\int }_{a^{\prime} }^{b}{P}_{n}(x){f}^{(n)}(x){\rm{d}}x,\end{eqnarray}$
with Pn being the periodic Bernoulli polynomial [6]. Dropping the remainder term Rn in equation (2. MEMF in Dimension 1
If a real function f is Cn on $[a,b]$ for some $n\in {{\mathbb{N}}}^{+}$ and a, $b\in {\mathbb{Z}}$ with $a\lt b$, then for given $m,p\in {\mathbb{N}}$ with $m\leqslant b-a$, the summation of f over integer points on $[a,b]$ can be evaluated by the formula
$\begin{eqnarray}\begin{array}{l}\displaystyle \sum _{i=a}^{b}f(i)=\displaystyle \sum _{i=a}^{a^{\prime} -1}f(i)+{\displaystyle \int }_{a^{\prime} }^{b}\displaystyle \frac{\sin ((2p+1)\pi x)}{\sin (\pi x)}f(x){\rm{d}}x\\ +\ \displaystyle \frac{f(a^{\prime} )+f(b)}{2}+\underset{r=1}{\overset{\rlap{-}n}{\displaystyle * \sum }}{\left(-1\right)}^{r}{T}_{2r,p}{M}_{2r-1}(a^{\prime} ,b)+{R}_{{mnp}},\end{array}\end{eqnarray}$
with $a^{\prime} =a+m$ and $\displaystyle {T}_{s,p}=\tfrac{2}{{\left(2\pi \right)}^{s}}\left(\zeta (s)-\displaystyle \sum _{k=1}^{p}\tfrac{1}{{k}^{s}}\right)$, where the remainder term is $\begin{eqnarray}{R}_{{mnp}}={\left(-1\right)}^{n}{\int }_{a^{\prime} }^{b}\displaystyle \sum _{| k| \gt p}\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}2\pi {kx}}}{{\left({\rm{i}}2\pi k\right)}^{n}}{f}^{(n)}(x){\rm{d}}x.\end{eqnarray}$
The $\zeta (s)$ in the definition of ${T}_{s,p}$ is the Riemann zeta function.Two modifications are made to the original Euler–Maclaurin formula. (i) We sum over the first m terms explicitly, which is trival but necessary for some quantum systems at low temperature. (ii) We keep part of the original remainder in our calculations to improve the accuracy of the summation.
We apply the Euler–Maclaurin formula to the summation on the left hand side after modification (i) and obtain
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \sum _{i=a}^{b}f(i) & = & \displaystyle \sum _{i=a}^{a^{\prime} -1}f(i)+{\displaystyle \int }_{a^{\prime} }^{b}f(x){\rm{d}}x+\displaystyle \frac{f(a^{\prime} )+f(b)}{2}\\ & & -\displaystyle \sum _{r=1}^{\rlap{-}n}\displaystyle \frac{{B}_{2r}}{(2r)!}{M}_{2r-1}(a^{\prime} ,b)+{R}_{{mn}},\end{array}\end{eqnarray}$
with $a^{\prime} =a+m$. The periodic Bernoulli polynomials of the remainder term in equation ( $\begin{eqnarray}{P}_{n}(x)=-n!\displaystyle \sum _{k\in Z-\{0\}}\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}2\pi {kx}}}{{\left({\rm{i}}2\pi k\right)}^{n}}.\end{eqnarray}$
Taking $2p$ terms from the remainder term with $| k| \leqslant p$ in equation ( $\begin{eqnarray*}\begin{array}{rcl}{R}_{{mn}} & = & {\left(-1\right)}^{n}{\displaystyle \int }_{a^{\prime} }^{b}\displaystyle \sum _{k\in Z-\{0\}}\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}2\pi {kx}}}{{\left({\rm{i}}2\pi k\right)}^{n}}{f}^{(n)}(x){\rm{d}}x\\ & = & {\left(-1\right)}^{n}{\displaystyle \int }_{a^{\prime} }^{b}\displaystyle \sum _{0\lt | k| \leqslant p}\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}2\pi {kx}}}{{\left({\rm{i}}2\pi k\right)}^{n}}{f}^{(n)}(x){\rm{d}}x+{R}_{{mnp}}.\end{array}\end{eqnarray*}$
Using integration by parts, we have $\begin{eqnarray}\begin{array}{rcl}{R}_{{mn}} & = & \displaystyle \sum _{l=0}^{n-1}\displaystyle \sum _{0\lt | k| \leqslant p}{\left(\displaystyle \frac{{\rm{i}}}{2\pi k}\right)}^{l+1}{M}_{l}(b,a^{\prime} )\\ & & +{\displaystyle \int }_{a^{\prime} }^{b}\displaystyle \sum _{0\lt | k| \leqslant p}{{\rm{e}}}^{{\rm{i}}2\pi {kx}}f(x){\rm{d}}x+{R}_{{mnp}}\\ & = & \displaystyle \sum _{r=1}^{n-}\displaystyle \sum _{0\lt k\leqslant p}\displaystyle \frac{2{\left(-1\right)}^{r+1}}{{\left(2\pi k\right)}^{2r}}{M}_{2r-1}(a^{\prime} ,b)\\ & & +{\displaystyle \int }_{a^{\prime} }^{b}\left(\displaystyle \frac{\sin ((2p+1)\pi x)}{\sin (\pi x)}-1\right)f(x){\rm{d}}x+{R}_{{mnp}}.\end{array}\end{eqnarray}$
Note that $\begin{eqnarray*}{B}_{2r}=2{\left(-1\right)}^{r+1}\displaystyle \frac{\zeta (2r)(2r)!}{{\left(2\pi \right)}^{2r}}.\end{eqnarray*}$
Substituting equation (Dropping the remainder term Rmnp in equation (4 ), an m-n-p cut of the MEMF is obtained, which can be used as an approximation of the summation on the left hand side. In practice, the integral in equation (4 ) can also be calculated by Obviously, similar conclusions can be drawn for $a^{\prime} =-\infty $, and for other results in this paper.
$\begin{eqnarray}{\int }_{a^{\prime} }^{b}f(x)\displaystyle \frac{\sin ((2p+1)\pi x)}{\sin (\pi x)}{\rm{d}}x=\displaystyle \sum _{k=-p}^{p}{\int }_{a^{\prime} }^{b}f(x){{\rm{e}}}^{{\rm{i}}2\pi {kx}}{\rm{d}}x,\end{eqnarray}$
whose right hand side is a summation of Fourier coefficients of f on $[a^{\prime} ,b]$. We remark that the integral kernel $\tfrac{\sin ((2p+1)\pi x)}{\sin (\pi x)}$ is identical to the multi-splits interference factor of an optical grating. If the conditions in theorem
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \sum _{i=a}^{+\infty }f(i) & = & \displaystyle \sum _{i=a}^{a^{\prime} -1}f(i)+{\displaystyle \int }_{a^{\prime} }^{+\infty }\displaystyle \frac{\sin ((2p+1)\pi x)}{\sin (\pi x)}f(x){\rm{d}}x\\ & & +\displaystyle \frac{f(a^{\prime} )}{2}+\displaystyle \sum _{r=1}^{\rlap{-}n}{\left(-1\right)}^{r}{T}_{2r,p}{M}_{2r-1}(a^{\prime} ,+\infty )+{R}_{{mnp}},\end{array}\end{eqnarray}$
with $\begin{eqnarray*}{R}_{{mnp}}={\left(-1\right)}^{n}{\int }_{a^{\prime} }^{+\infty }\displaystyle \sum _{| k| \gt p}\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}2\pi {kx}}}{{\left({\rm{i}}2\pi k\right)}^{n}}{f}^{(n)}(x){\rm{d}}x.\end{eqnarray*}$
We reveal the following connections to the Euler–Maclaurin formula and to the Poisson summation formula.
| 1.Taking $m=0,p=0$ in theorem | |
| 2.Taking $m=0,n=1$ in theorem $\begin{eqnarray*}\displaystyle \sum _{i=a}^{b}f(i)=\displaystyle \sum _{k=-\infty }^{+\infty }{\int }_{a}^{b}f(x){{\rm{e}}}^{{\rm{i}}2\pi {kx}}{\rm{d}}x+\displaystyle \frac{f(a)+f(b)}{2}.\end{eqnarray*}$ Furthermore, if all the terms in the above equation are finite and the summation on the left hand side converges as $a\to -\infty $ and $b\to +\infty $, then it becomes the Poisson summation formula $\begin{eqnarray*}\displaystyle \sum _{i=-\infty }^{+\infty }f(i)=\displaystyle \sum _{k=-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }f(x){{\rm{e}}}^{{\rm{i}}2\pi {kx}}{\rm{d}}x.\end{eqnarray*}$ |
If the conditions in theorem
| 1. $\begin{eqnarray*}| {R}_{{mnp}}| \leqslant {R}_{{mnp}}^{A}\equiv {T}_{n,p}{\int }_{a^{\prime} }^{b}\left|{f}^{(n)}(x)\right|{\rm{d}}x.\end{eqnarray*}$ | |
| 2.If n is odd, and ${f}^{(n)}(x)$ is monotone with respect to x on $[a^{\prime} ,b]$, then $\begin{eqnarray*}\left|{R}_{{mnp}}\right|\leqslant {R}_{{mnp}}^{B}\equiv 2{T}_{n+1,p}\left|{M}_{n}(a^{\prime} ,b)\right|.\end{eqnarray*}$ If n is even, and ${f}^{(n)}(x)$ is monotone with respect to x on $[a^{\prime} ,b]$, then $\begin{eqnarray*}\left|{R}_{{mnp}}\right|\leqslant {R}_{{mnp}}^{C}\equiv {T}_{n+1,p}\left|{M}_{n}(a^{\prime} ,b)\right|.\end{eqnarray*}$ |
| 1.Compute $\begin{eqnarray*}\begin{array}{l}\left|{R}_{{mnp}}\right|=\left|{\left(-1\right)}^{n}{\displaystyle \int }_{a^{\prime} }^{b}\displaystyle \sum _{| k| \gt p}\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}2\pi {kx}}}{{\left({\rm{i}}2\pi k\right)}^{n}}{f}^{(n)}(x){\rm{d}}x\right|\leqslant {\displaystyle \int }_{a^{\prime} }^{b}\displaystyle \sum _{| k| \gt p}\left|\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}2\pi {kx}}}{{\left({\rm{i}}2\pi k\right)}^{n}}{f}^{(n)}(x)\right|{\rm{d}}x\\ \qquad \quad={T}_{n,p}{\displaystyle \int }_{a^{\prime} }^{b}\left|{f}^{(n)}(x)\right|{\rm{d}}x.\end{array}\end{eqnarray*}$ | |
| 2.For odd n, we estimate $\begin{eqnarray*}\begin{array}{l}\left|{R}_{{mnp}}\right|=\left|{\left(-1\right)}^{n}{\displaystyle \int }_{a^{\prime} }^{b}\displaystyle \sum _{| k| \gt p}\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}2\pi {kx}}}{{\left({\rm{i}}2\pi k\right)}^{n}}{f}^{(n)}(x){\rm{d}}x\right|\\ \quad \leqslant \displaystyle \frac{1}{{\left(2\pi \right)}^{n}}\displaystyle \sum _{k\gt p}\displaystyle \frac{2}{{k}^{n}}\left|{\displaystyle \int }_{a^{\prime} }^{b}\sin (2\pi {kx}){f}^{(n)}(x){\rm{d}}x\right|\\ =\ \displaystyle \frac{1}{{\left(2\pi \right)}^{n}}\displaystyle \sum _{k\gt p}\displaystyle \frac{2}{{k}^{n}}\ \left|\displaystyle \sum _{j=1}^{2k(b-a^{\prime} )}{\displaystyle \int }_{a^{\prime} +\tfrac{j-1}{2k}}^{a^{\prime} +\tfrac{j}{2k}}\sin (2\pi {kx})\left({f}^{(n)}(x)-{f}^{(n)}(b)\right){\rm{d}}x\right|.\end{array}\end{eqnarray*}$ Note that the summation is an alternating summation, we have $\begin{eqnarray*}\begin{array}{l}\left|{R}_{{mnp}}\right|\leqslant \displaystyle \frac{1}{{\left(2\pi \right)}^{n}}\displaystyle \sum _{k\gt p}\displaystyle \frac{2}{{k}^{n}}\left|{\displaystyle \int }_{a^{\prime} }^{a^{\prime} +\tfrac{1}{2k}}\sin (2\pi {kx})\left({f}^{(n)}(x)-{f}^{(n)}(b)\right){\rm{d}}x\right|\\ \,\leqslant \,\displaystyle \frac{1}{{\left(2\pi \right)}^{n}}\displaystyle \sum _{k\gt p}\displaystyle \frac{2}{{k}^{n}}\left|{f}^{(n)}(a^{\prime} )-{f}^{(n)}(b)\right|\displaystyle \frac{1}{\pi k}\\ \,=\ 2{T}_{n+1,p}\left|{M}_{n}(a^{\prime} ,b)\right|.\end{array}\end{eqnarray*}$ Similarly, for even n, we estimate $\begin{eqnarray*}\begin{array}{l}\left|{R}_{{mnp}}\right|=\left|{\left(-1\right)}^{n}{\displaystyle \int }_{a^{\prime} }^{b}\displaystyle \sum _{| k| \gt p}\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}2\pi {kx}}}{{\left({\rm{i}}2\pi k\right)}^{n}}{f}^{(n)}(x){\rm{d}}x\right|\\ \,\leqslant \,\displaystyle \frac{1}{{\left(2\pi \right)}^{n}}\displaystyle \sum _{k\gt p}\displaystyle \frac{2}{{k}^{n}}\left|{\displaystyle \int }_{a^{\prime} }^{b}\cos (2\pi {kx}){f}^{(n)}(x){\rm{d}}x\right|\\ \,=\ \displaystyle \frac{1}{{\left(2\pi \right)}^{n}}\displaystyle \sum _{k\gt p}\displaystyle \frac{2}{{k}^{n}}\left|\left({\displaystyle \int }_{a^{\prime} }^{a^{\prime} +\tfrac{1}{4k}}+\displaystyle \sum _{j=1}^{2k(b-a^{\prime} )-1}{\displaystyle \int }_{a^{\prime} +\tfrac{j-1/2}{2k}}^{a^{\prime} +\tfrac{j+1/2}{2k}}+{\displaystyle \int }_{b-\tfrac{1}{4k}}^{b}\right)\\ \times \cos (2\pi {kx})\left({f}^{(n)}(x)-{f}^{(n)}(b)\right){\rm{d}}x\right|\\ \,\leqslant \,\displaystyle \frac{1}{{\left(2\pi \right)}^{n}}\displaystyle \sum _{k\gt p}\displaystyle \frac{2}{{k}^{n}}{\displaystyle \int }_{a^{\prime} }^{a^{\prime} +\tfrac{1}{4k}}\cos (2\pi {kx})\left|{f}^{(n)}(x)-{f}^{(n)}(b)\right|{\rm{d}}x\\ \,\leqslant \,\displaystyle \frac{1}{{\left(2\pi \right)}^{n}}\displaystyle \sum _{k\gt p}\displaystyle \frac{2}{{k}^{n}}\left|{f}^{(n)}(a^{\prime} )-{f}^{(n)}(b)\right|\displaystyle \frac{1}{2\pi k}\\ \,=\,\ {T}_{n+1,p}\left|{M}_{n}(a^{\prime} ,b)\right|.\end{array}\end{eqnarray*}$ |
If the conditions in theorem
$\begin{eqnarray*}\left|{R}_{{mnp}}\right|\leqslant {\tilde{R}}_{{mnp}}^{A}\equiv {T}_{n,p}{N}_{n-1}{\rm{\Delta }}{M}_{n-1},\end{eqnarray*}$
with ${\rm{\Delta }}{M}_{n-1}={M}_{\max }-{M}_{\min }$, where Mmax and Mmin are the maximum and minimum values of ${f}^{(n-1)}$ on $[a^{\prime} ,b]$, respectively.Note that in theorem 1 , the behavior of divergence is usually kept as n → + ∞ with fixed m and p, which only postpones the divergence. On the other hand, it converges as m → + ∞ or p → + ∞ with fixed n, which corresponds to the definition of the left hand side and a Fourier expansion of the original remainder term, respectively. In short, n is for efficiency, and m and p are for convergency.
When the MEMF is used to work out a summation approximately, one first set a tolerance of error R*, and then choose some m and n according to the difficulty on calculating the right hand side other than the remainder term of equation (4 ). Next he estimates ${\int }_{a^{\prime} }^{b}\left|{f}^{(n)}(x)\right|{\rm{d}}x$, ${M}_{n}(a^{\prime} ,b)$, or both Nn−1 and ΔMn−1, and then choose a p according to theorem 2 or corollary 2 to suppress ∣Rmnp∣ by the Tn,p or Tn+1,p term, so that it is not more than R*. Finally the right hand side other than Rmnp is calculated as an approximation of the left hand side with an satisfied error.
3. Generalization to Dimension 2
Let D be the rectangle
$\begin{eqnarray*}D=\{(x,y)\in {{\mathbb{R}}}^{2}| a\leqslant x\lt b,c\leqslant y\lt d\},\end{eqnarray*}$
illustrated as in the left panel of figure 1, where $a,b,c$, and d are integers satisfying $a\lt b$, $c\lt d$. If a function $f:\bar{D}\to {\mathbb{R}}$ with $\bar{D}=D\cup \partial D$ is ${C}^{n+n^{\prime} }$ for some positive integers $n,n^{\prime} $, then for given non-negative integers $m,m^{\prime} ,p,p^{\prime} $, the summation of $f(x,y)$ on all integer pairs $(i,j)\in D$ can be expressed in the following form $\begin{eqnarray}\displaystyle \sum _{\mathrm{integer}\ \mathrm{pairs}\ (i,j)\in D}f(i,j)=A+L+V+R,\end{eqnarray}$
with $\begin{eqnarray}A={\int }_{D}{\rm{d}}x{\rm{d}}y\ {a}_{p}^{x}{a}_{p^{\prime} }^{y}f(x,y),\end{eqnarray}$
$\begin{eqnarray}L={\int }_{\partial D}-{a}_{p}^{x}{B}_{n^{\prime} ,p^{\prime} }^{y}f(x,y){\rm{d}}x+{a}_{p^{\prime} }^{y}{B}_{n,p}^{x}f(x,y){\rm{d}}y,\end{eqnarray}$
$\begin{eqnarray}V=\displaystyle \sum _{(i,j)\in \mathrm{vertices}\ \mathrm{of}\ D}{s}_{i,j}{B}_{n,p}^{x}{B}_{n^{\prime} ,p^{\prime} }^{y}f(i,j),\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}R={R}_{{np},n^{\prime} p^{\prime} }\\ =\ {\int }_{D}{\rm{d}}x{\rm{d}}y\ \left(\left({a}_{p}^{x}+{b}_{n,p}^{x}\right){c}_{n^{\prime} ,p^{\prime} }^{y}+\left({a}_{p^{\prime} }^{y}+{b}_{n^{\prime} ,p^{\prime} }^{y}\right){c}_{n,p}^{x}+{c}_{n,p}^{x}{c}_{n^{\prime} ,p^{\prime} }^{y}\right)f(x,y),\end{array}\end{eqnarray}$
and the linear operators $\begin{eqnarray}{B}_{n,p}^{x}=-\displaystyle \frac{1}{2}+\displaystyle \sum _{r=1}^{\rlap{-}n}{\left(-1\right)}^{r+1}{T}_{2r,p}\displaystyle \frac{{\partial }^{2r-1}}{\partial {x}^{2r-1}},\end{eqnarray}$
$\begin{eqnarray}{a}_{p}^{x}=\displaystyle \frac{\sin \left((2p+1)\pi x\right)}{\sin (\pi x)},\end{eqnarray}$
$\begin{eqnarray}{b}_{n,p}^{x}=\displaystyle \frac{\partial }{\partial x}{B}_{n,p}^{x}=-\displaystyle \frac{1}{2}\displaystyle \frac{{\rm{d}}}{{\rm{d}}x}+\displaystyle \sum _{r=1}^{n-}{\left(-1\right)}^{r+1}{T}_{2r,p}\displaystyle \frac{{\partial }^{2r}}{\partial {x}^{2r}},\end{eqnarray}$
$\begin{eqnarray}{c}_{n,p}^{x}={\left(-1\right)}^{n}\left(\displaystyle \sum _{| k| \gt p}\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}2\pi {kx}}}{{\left({\rm{i}}2\pi k\right)}^{n}}\right)\displaystyle \frac{{\partial }^{n}}{\partial {x}^{n}}.\end{eqnarray}$
The operators with superscript y are defined similarly. The factor ${s}_{i,j}$ in equation (We rewrite equation (
$\begin{eqnarray}\displaystyle \sum _{i=a}^{b-1}f(i)={\int }_{a}^{b}\left({a}_{p}^{x}+{b}_{n,p}^{x}\right)f(x){\rm{d}}x+{\int }_{a}^{b}{c}_{n,p}^{x}f(x){\rm{d}}x.\end{eqnarray}$
Applying the formula in equation ( $\begin{eqnarray*}\begin{array}{rcl}\displaystyle \sum _{i=a}^{b-1}\displaystyle \sum _{j=c}^{d-1}f(i,j) & = & {\displaystyle \int }_{D}\left({a}_{p}^{x}+{b}_{n,p}^{x}\right)\left({a}_{p^{\prime} }^{y}+{b}_{n^{\prime} ,p^{\prime} }^{y}\right)f(x,y){\rm{d}}x{\rm{d}}y+{R}_{{np},n^{\prime} p^{\prime} }\\ & = & {\displaystyle \int }_{D}{a}_{p}^{x}{a}_{p^{\prime} }^{y}f(x,y){\rm{d}}x{\rm{d}}y+{\displaystyle \int }_{D}\left({a}_{p}^{x}{b}_{n^{\prime} ,p^{\prime} }^{y}+{b}_{n,p}^{x}{a}_{p^{\prime} }^{y}\right)f(x,y){\rm{d}}x{\rm{d}}y\\ & & +{\displaystyle \int }_{D}{b}_{n,p}^{x}{b}_{n^{\prime} ,p^{\prime} }^{y}f(x,y){\rm{d}}x{\rm{d}}y+{R}_{{np},n^{\prime} p^{\prime} }\\ & = & A+{\displaystyle \int }_{\partial D}\left(-{a}_{p}^{x}{B}_{n^{\prime} ,p^{\prime} }^{y}f(x,y){\rm{d}}x+{B}_{n,p}^{x}{a}_{p^{\prime} }^{y}f(x,y){\rm{d}}y\right)\\ & & +{\displaystyle \int }_{D}\displaystyle \frac{\partial }{\partial x}\displaystyle \frac{\partial }{\partial y}{B}_{n,p}^{x}{B}_{n^{\prime} ,p^{\prime} }^{y}f(x,y){\rm{d}}x{\rm{d}}y+R\\ & = & A+L+V+R.\end{array}\end{eqnarray*}$
That is equation (Note that (i) equation (
Suppose P is a set of integer pairs with ${P}_{0}\subset P$, ${P}_{1}=P-{P}_{0}$, and
$\begin{eqnarray}D=\bigcup _{(i,j)\in {P}_{1}}\{(x,y)\in {{\mathbb{R}}}^{2}| i\leqslant x\lt i+1,j\leqslant y\lt j+1\}.\end{eqnarray}$
Under the conditions of lemma $\begin{eqnarray}\displaystyle \sum _{(i,j)\in P}f(i,j)=\displaystyle \sum _{(i,j)\in {P}_{0}}f(i,j)+A+L+V+R,\end{eqnarray}$
if all the terms are finite and well-defined, where A, L, V, and R are defined by equations (Again, dropping the remainder term R, an approximation of the original summation in Dimension 2 is obtained.
If f satisfies the conditions in theorem
$\begin{eqnarray*}\begin{array}{l}| {R}_{{np},n^{\prime} p^{\prime} }| \leqslant {R}_{{np},n^{\prime} p^{\prime} }^{A}\equiv {T}_{n,p}\left((2p^{\prime} +1){\displaystyle \int }_{D}\left|{f}_{n,0}(x,y)\right|{\rm{d}}x{\rm{d}}y+\displaystyle \frac{1}{2}{\displaystyle \int }_{\partial D}\left|{f}_{n,0}(x,y){\rm{d}}x\right|\right)\\ +\,{T}_{n^{\prime} ,p^{\prime} }\left((2p+1){\displaystyle \int }_{D}\ \left|{f}_{0,n^{\prime} }(x,y)\right|{\rm{d}}x{\rm{d}}y+\displaystyle \frac{1}{2}{\displaystyle \int }_{\partial D}\left|{f}_{0,n^{\prime} }(x,y){\rm{d}}y\right|\right)\\ +\,{T}_{n,p}\displaystyle \sum _{r=1}^{n-^{\prime} }{T}_{2r,p^{\prime} }{\displaystyle \int }_{\partial D}\left|{f}_{n,2r-1}(x,y){\rm{d}}x\right|+{T}_{n^{\prime} ,p^{\prime} }\displaystyle \sum _{r=1}^{\rlap{-}n}{T}_{2r,p}{\displaystyle \int }_{\partial D}\left|{f}_{2r-1,n^{\prime} }(x,y){\rm{d}}y\right|\\ +\,{T}_{n,p}{T}_{n^{\prime} ,p^{\prime} }{\displaystyle \int }_{D}\ \left|{f}_{n,n^{\prime} }(x,y)\right|{\rm{d}}x{\rm{d}}y\end{array}\end{eqnarray*}$
with ${f}_{\mu ,\nu }(x,y)\equiv \tfrac{{\partial }^{\mu +\nu }}{\partial {x}^{\mu }\partial {y}^{\nu }}f(x,y)$.Equation (
$\begin{eqnarray}\begin{array}{l}\left|{R}_{{np},n^{\prime} p^{\prime} }\right|=\left|{\displaystyle \int }_{D}\ \left(\left({a}_{p^{\prime} }^{y}+{b}_{n^{\prime} ,p^{\prime} }^{y}\right){c}_{n,p}^{x}+\left({a}_{p}^{x}+{b}_{n,p}^{x}\right){c}_{n^{\prime} ,p^{\prime} }^{y}+{c}_{n,p}^{x}{c}_{n^{\prime} ,p^{\prime} }^{y}\right)f(x,y){\rm{d}}x{\rm{d}}y\right|\\ \leqslant \ {\displaystyle \int }_{D}\left(\left|{a}_{p^{\prime} }^{y}{c}_{n,p}^{x}f(x,y)\right|+\left|{a}_{p}^{x}{c}_{n^{\prime} ,p^{\prime} }^{y}f(x,y)\right|+\left|{c}_{n,p}^{x}{c}_{n^{\prime} ,p^{\prime} }^{y}f(x,y)\right|\right){\rm{d}}x{\rm{d}}y\ \\ +\ \left|{\displaystyle \int }_{D}\ \left({b}_{n^{\prime} ,p^{\prime} }^{y}{c}_{n,p}^{x}f(x,y)+{b}_{n,p}^{x}{c}_{n^{\prime} ,p^{\prime} }^{y}f(x,y)\right){\rm{d}}x{\rm{d}}y\right|.\end{array}\end{eqnarray}$
Note that, following the procedure in the proof of theorem $\begin{eqnarray}\left|{c}_{n,p}^{x}f(x,y)\right|\leqslant {T}_{n,p}\left|{f}_{n,0}(x,y)\right|,\end{eqnarray}$
$\begin{eqnarray}\left|{c}_{n^{\prime} ,p^{\prime} }^{y}f(x,y)\right|\leqslant {T}_{n^{\prime} ,p^{\prime} }\left|{f}_{0,n^{\prime} }(x,y)\right|.\end{eqnarray}$
Thus, $\begin{eqnarray*}\begin{array}{l}{\int }_{D}\left(\left|{a}_{p\mbox{'}}^{y}{c}_{n,p}^{x}f(x,y)\right|+\left|{a}_{p}^{x}{c}_{n\mbox{'},p\mbox{'}}^{y}f(x,y)\right|+\left|{c}_{n,p}^{x}{c}_{n\mbox{'},p\mbox{'}}^{y}f(x,y)\right|\right){\rm{d}}x{\rm{d}}y\\ \quad \leqslant \,{T}_{n,p}{\int }_{D}\left|{a}_{p\mbox{'}}^{y}{f}_{n,0}(x,y)\right|{\rm{d}}x{\rm{d}}y+{T}_{n\mbox{'},p\mbox{'}}{\int }_{D}\,\left|{a}_{p}^{x}{f}_{0,n\mbox{'}}(x,y)\right|{\rm{d}}x{\rm{d}}y\,+\,{T}_{n,p}{T}_{n\mbox{'},p\mbox{'}}{\int }_{D}\ \left|{f}_{n,n\mbox{'}}(x,y)\right|{\rm{d}}x{\rm{d}}y.\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\left|{\displaystyle \int }_{D}\ \left({b}_{n^{\prime} ,p^{\prime} }^{y}{c}_{n,p}^{x}f(x,y)+{b}_{n,p}^{x}{c}_{n^{\prime} ,p^{\prime} }^{y}f(x,y)\right){\rm{d}}x{\rm{d}}y\right|\\ \quad =\,\left|{\displaystyle \int }_{D}{\rm{d}}\left(-{B}_{n^{\prime} ,p^{\prime} }^{y}{c}_{n,p}^{x}f(x,y){\rm{d}}x+{B}_{n,p}^{x}{c}_{n^{\prime} ,p^{\prime} }^{y}f(x,y){\rm{d}}y\right)\right|\\ \quad =\,\left|{\displaystyle \int }_{\partial D}\left(-{B}_{n^{\prime} ,p^{\prime} }^{y}{c}_{n,p}^{x}f(x,y){\rm{d}}x+{B}_{n,p}^{x}{c}_{n^{\prime} ,p^{\prime} }^{y}f(x,y){\rm{d}}y\right)\right|\\ \quad \leqslant \,{T}_{n,p}{\displaystyle \int }_{\partial D}\left|{B}_{n^{\prime} ,p^{\prime} }^{y}{f}_{n,0}(x,y){\rm{d}}x\right|+{T}_{n^{\prime} ,p^{\prime} }{\displaystyle \int }_{\partial D}\left|{B}_{n,p}^{x}{f}_{0,n^{\prime} }(x,y){\rm{d}}y\right|.\end{array}\end{eqnarray*}$
Note that $\begin{eqnarray*}\begin{array}{rcl}\left|{a}_{p}^{x}\right| & \leqslant & 2p+1,\\ \left|{B}_{n,p}^{x}f(x,y)\right| & = & \left|\left(-\displaystyle \frac{1}{2}+\displaystyle \sum _{r=1}^{\rlap{-}n}{\left(-1\right)}^{r+1}{T}_{2r,p}\displaystyle \frac{{\partial }^{2r-1}}{\partial {x}^{2r-1}}\right)f(x,y)\right|\\ & \leqslant & \displaystyle \frac{1}{2}\left|f(x,y)\right|+\displaystyle \sum _{r=1}^{\rlap{-}n}{T}_{2r,p}\left|{f}_{2r-\mathrm{1,0}}(x,y)\right|.\end{array}\end{eqnarray*}$
Similarly, $\begin{eqnarray*}\begin{array}{rcl}\left|{a}_{p^{\prime} }^{y}\right| & \leqslant & 2p^{\prime} +1,\\ \left|{B}_{n^{\prime} ,p^{\prime} }^{y}f(x,y)\right| & = & \left|\left(-\displaystyle \frac{1}{2}+\displaystyle \sum _{r=1}^{n-^{\prime} }{\left(-1\right)}^{r+1}{T}_{2r,p^{\prime} }\displaystyle \frac{{\partial }^{2r-1}}{\partial {y}^{2r-1}}\right)f(x,y)\right|\\ & \leqslant & \displaystyle \frac{1}{2}\left|f(x,y)\right|+\displaystyle \sum _{r=1}^{n-^{\prime} }{T}_{2r,p^{\prime} }\left|{f}_{\mathrm{0,2}r-1}(x,y)\right|.\end{array}\end{eqnarray*}$
Thus theorem 4 holds. Obviously, for given n, $n^{\prime} $ and f, one can always choose proper p and $p^{\prime} $ to achieve as high accuracy as expected. For $n^{\prime} =n$ and $p^{\prime} =p$, the formula is simpler as follows.
$n=n^{\prime} $ and $p=p^{\prime} $. If f satisfies the conditions in theorem
$\begin{eqnarray*}\begin{array}{rcl}| {R}_{{np},{np}}| & \leqslant & {R}_{{np},{np}}^{A}\,=\,{T}_{n,p}\left((2p+1){\int }_{D}\left(\left|{f}_{n,0}(x,y)\right|+\left|{f}_{0,n}(x,y)\right|\right){\rm{d}}x{\rm{d}}y\right.\\ & & +\ \left.\frac{1}{2}{\int }_{\partial D}\left|{f}_{0,n}(x,y){\rm{d}}y\right|+\left|{f}_{n,0}(x,y){\rm{d}}x\right|\right)\,+{T}_{n,p}\underset{r=1}{\overset{\rlap{-}n}{}}{T}_{2r,p}\left({\int }_{\partial D}\left|{f}_{n,2r-1}(x,y){\rm{d}}x\right|\right.\\ & & +\ \left.\left|{f}_{2r-1,n}(x,y){\rm{d}}y\right|,\right)+{T}_{n,p}^{2}{\int }_{D}\ \left|{f}_{n,n}(x,y)\right|{\rm{d}}x{\rm{d}}y.\end{array}\end{eqnarray*}$
Figure 1. Summation points (left) in lemma |
4. Applications
In this part, we give approximate expressions of partition functions in some simple systems by using the MEMF.
4.1. Partition function of a one-dimensional infinite square well
4.1.1. Partition function
For a one-dimensional infinite square well, the eigen energy is10 ) gives
$\begin{eqnarray*}{\epsilon }_{l}=\epsilon {l}^{2},\qquad l\in {{\mathbb{N}}}^{+},\end{eqnarray*}$
where ε = ε1 is the eigen energy of the ground state. Thus the partition function $\begin{eqnarray*}Z=\displaystyle \sum _{l=1}^{+\infty }f(l),\end{eqnarray*}$
with $f(x)={{\rm{e}}}^{-{{Bx}}^{2}}$ and B = βε1. Then equation ( $\begin{eqnarray*}Z={Z}_{m}+{W}_{{np}}(m+1)+{R}_{{mnp}},\end{eqnarray*}$
with $\begin{eqnarray}{Z}_{m}=\displaystyle \sum _{l=1}^{m}{{\rm{e}}}^{-{{Bl}}^{2}},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{W}_{{np}}(x) & = & {T}_{p}(x)+{U}_{{np}}(x),\\ {T}_{p}(x) & = & {\displaystyle \int }_{x}^{+\infty }\displaystyle \frac{\sin ((2p+1)\pi \xi )}{\sin (\pi \xi )}f(\xi ){\rm{d}}\xi \,=\,\displaystyle \frac{1}{2}\sqrt{\displaystyle \frac{\pi }{B}}\displaystyle \sum _{k=-p}^{p}{{\rm{e}}}^{-\tfrac{{\pi }^{2}{k}^{2}}{B}}\mathrm{erfc}\left(\sqrt{B}x-\displaystyle \frac{{\rm{i}}\pi k}{\sqrt{B}}\right),\\ {U}_{{np}}(x) & = & \displaystyle \frac{1}{2}f(x)+\displaystyle \sum _{r=1}^{\rlap{-}n}{\left(-1\right)}^{r}{T}_{2r,p}{f}^{(2r-1)}(x)\,=\,\displaystyle \frac{1}{2}{{\rm{e}}}^{-{{Bx}}^{2}}+\displaystyle \sum _{r=1}^{\rlap{-}n}{\left(-1\right)}^{r+1}{T}_{2r,p}{B}^{r-\tfrac{1}{2}}{H}_{2r-1}(\sqrt{B}x){{\rm{e}}}^{-{{Bx}}^{2}},\end{array}\end{eqnarray}$
where $\mathrm{erfc}(x)=\tfrac{2}{\sqrt{\pi }}{\int }_{x}^{+\infty }{{\rm{e}}}^{-{z}^{2}}{\rm{d}}z$ is the complementary error function, and ${H}_{n}(x)={\left(-1\right)}^{n}{{\rm{e}}}^{{x}^{2}}\tfrac{{{\rm{d}}}^{n}}{{\rm{d}}{x}^{n}}{{\rm{e}}}^{-{x}^{2}}$ is the Hermite polynomial.4.1.2. Remainder estimation
It can be proved with the help of the integral representations of Hermite polynomials [12] that2 by
$\begin{eqnarray*}\left|{H}_{n}(x){{\rm{e}}}^{-{x}^{2}}\right|\leqslant \displaystyle \frac{{2}^{n}{\rm{\Gamma }}(\tfrac{n+1}{2})}{\sqrt{\pi }}\end{eqnarray*}$
for all $x\in {\mathbb{R}}$ and all $n\in {\mathbb{N}}$, and ${H}_{n}(x){{\rm{e}}}^{-{x}^{2}}$ has at most $\left[\tfrac{n+3}{2}\right]$ monotonic intervals on [m +1, +∞), the remainder term can be controlled according to corollary $\begin{eqnarray}\left|{R}_{{mnp}}\right|\leqslant {\tilde{R}}_{{mnp}}^{A}\leqslant {\bar{R}}_{{np}}^{A}\equiv {T}_{n,p}\displaystyle \frac{\left(n\,-+1\right){2}^{n}{\rm{\Gamma }}(\tfrac{n}{2})}{\sqrt{\pi }}{B}^{\tfrac{n-1}{2}}\end{eqnarray}$
for n > 1.To have a better estimation of the remainder term, especially when m is large, we need a better estimation of Hn(x). Note that ${H}_{n}(x){{\rm{e}}}^{-{x}^{2}/2}$ is the (unnormalized) wave function of a quantum oscillator. By considering the classical corresponding of a quantum oscillator [13], we have the following conjecture
An upper bound of Hermite polynomials can be estimated by
$\begin{eqnarray}\left|{H}_{n}(x){{\rm{e}}}^{-{x}^{2}}\right|\leqslant \displaystyle \frac{{2}^{n}{\rm{\Gamma }}(\tfrac{n+1}{2})}{\sqrt{\pi }}{g}_{n}({\rm{\min }}(x,{x}_{n})){{\rm{e}}}^{-{x}^{2}/2},\end{eqnarray}$
with $\begin{eqnarray*}{g}_{n}(x)={\left(1-\displaystyle \frac{{x}^{2}}{2n+1}\right)}^{-1/2},\,\end{eqnarray*}$
and ${x}_{n}=\sqrt{2n+1}\left(1-\tfrac{\pi }{2(2n+1)}\right)$ for all $n\in {{\mathbb{N}}}^{+}$, and $x\gt 0$.We have checked numerically that the conjecture holds at least for n ≤ 20, and one can easily check given reasonable larger n numerically when necessary. As an illustration, the first 9 values of xn and gn(xn) are listed in table 1. If equation (29 ) holds, there is a rougher but simpler estimation:2 by30 ) which is based on Conjecture 1 on Hermite polynomials. From equation (32 ), it can be seen that when n is properly large, the remainder will be controlled for small B due to the power term, and when m is properly large, the remainder will be controlled for large B due to the exponential term, and when p is properly large, the remainder will be suppressed overall due to the effect of the Tn,p term. Therefore, one can easily make the remainder term as small as possible. We list some values of ${\bar{R}}_{{np}}^{A}$ (defined in equation (28 )), ${\tilde{R}}_{0{np}}^{H}$ [defined in equation (32 )] in table 2 and 3, respectively. It can be seen that the result can be as accurate as possible by choosing proper m, n, and p. If we regard ${\tilde{R}}_{{mnp}}^{H}$ as a function of B, the maximum is achieved at ${B}_{m}=(n-1)/{\left(m+1\right)}^{2}$, so that the remainder term can be controlled for all values of B. We plot ${\tilde{R}}_{252}^{H}$ as a function of B as an example in figure 2, where Bm = 4/9. Therefore we have33 ), only the term Tn,p depends on p, and it decreases to 0 as p → +∞. Thus with increasing p and fixed m and n, Zm + Wnp(m + 1) is a series of functions with respect to B that converges uniformly to Z as a function with respect to B. Equation (29 ) can further be improved for large x, and we leave it for future work.
$\begin{eqnarray}\left|{H}_{n}(x){{\rm{e}}}^{-{x}^{2}}\right|\leqslant {G}_{n}{{\rm{e}}}^{-{x}^{2}/2},\end{eqnarray}$
with $\begin{eqnarray}{G}_{n}\equiv \displaystyle \frac{{2}^{n}{\rm{\Gamma }}(\tfrac{n+1}{2})}{\sqrt{\pi }}{g}_{n}({x}_{n})=\displaystyle \frac{{2}^{n+1}(2n+1){\rm{\Gamma }}(\tfrac{n+1}{2})}{\pi \sqrt{4(2n+1)-\pi }},\end{eqnarray}$
for all $n\in {{\mathbb{N}}}^{+}$, since gn(x) increases as x increases at x > 0, where we have inserted $\begin{eqnarray*}\begin{array}{l}{g}_{n}({x}_{n})={\left(1-{\left(1-\displaystyle \frac{\pi }{2(2n+1)}\right)}^{2}\right)}^{-1/2}=\ {\left(\displaystyle \frac{2\pi }{2(2n+1)}\right)}^{-1/2}{\left(1-\displaystyle \frac{\pi }{4(2n+1)}\right)}^{-1/2}.\end{array}\end{eqnarray*}$
Then the remainder can be controlled according to corollary $\begin{eqnarray}\left|{R}_{{mnp}}\right|\leqslant {\tilde{R}}_{{mnp}}^{H}\equiv {T}_{n,p}2(n\,-+1){G}_{n-1}{{\rm{e}}}^{-\tfrac{B{\left(m+1\right)}^{2}}{2}}{B}^{\tfrac{n-1}{2}}\end{eqnarray}$
for n > 1. The superscript H here and in the following implies that we have applied equation ( $\begin{eqnarray}| {R}_{{mnp}}| \leqslant {T}_{n,p}2(n\,-+1){G}_{n-1}{{\rm{e}}}^{-\tfrac{n-1}{2}}\displaystyle \frac{{\left(n-1\right)}^{\tfrac{n-1}{2}}}{{\left(m+1\right)}^{n-1}},\end{eqnarray}$
for all values of B. On the right hand side of equation (
Figure 2. The error bound ${\tilde{R}}_{252}^{H}$ [defined in equation ( |
Figure 3. Summation region of the partition function of a two-dimensional square well. |
Table 1. Approximate values of the first several xn and gn(xn). |
| n | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|
| xn | 1.534 | 2.052 | 2.476 | 2.843 | 3.170 | 3.467 | 3.742 | 3.999 | 4.240 |
| gn(xn) | 1.374 | 1.584 | 1.772 | 1.942 | 2.099 | 2.245 | 2.382 | 2.512 | 2.635 |
Table 2. Approximate values of ${\bar{R}}_{{np}}^{A}$ [defined in equation ( |
| n | 3 | 5 | 7 | 9 | 11 | 13 | 15 |
|---|---|---|---|---|---|---|---|
| p | |||||||
| 0 | 7.8 × 10−2 | 1.5 × 10−2 | 5.0 × 10−3 | 2.2 × 10−3 | 1.2 × 10−3 | 7.8 × 10−4 | 5.9 × 10−4 |
| 1 | 1.3 × 10−2 | 5.4 × 10−4 | 4.1 × 10−5 | 4.4 × 10−6 | 6.0 × 10−7 | 9.6 × 10−8 | 1.8 × 10−8 |
| 2 | 5.0 × 10−3 | 8.3 × 10−5 | 2.7 × 10−6 | 1.2 × 10−7 | 7.1 × 10−9 | 5.0 × 10−10 | 4.2 × 10−11 |
| 3 | 2.6 × 10−3 | 2.3 × 10−5 | 3.9 × 10−7 | 9.8 × 10−9 | 3.2 × 10−10 | 1.2 × 10−11 | 5.7 × 10−13 |
| 4 | 1.6 × 10−3 | 8.6 × 10−6 | 9.2 × 10−8 | 1.4 × 10−9 | 2.9 × 10−11 | 7.1 × 10−13 | 2.1 × 10−14 |
Table 3. Approximate values of ${\tilde{R}}_{0{np}}^{H}$ [defined in equation ( |
| n | 3 | 5 | 7 | 9 | 11 | 13 | 15 |
|---|---|---|---|---|---|---|---|
| p | |||||||
| 0 | 6.5 × 10−2 | 1.6 × 10−2 | 6.4 × 10−3 | 3.2 × 10−3 | 1.9 × 10−3 | 1.4 × 10−3 | 1.1 × 10−3 |
| 1 | 1.1 × 10−2 | 5.8 × 10−4 | 5.3 × 10−5 | 6.4 × 10−6 | 9.5 × 10−7 | 1.7 × 10−7 | 3.4 × 10−8 |
| 2 | 4.1 × 10−3 | 9. × 10−5 | 3.4 × 10−6 | 1.8 × 10−7 | 1.1 × 10−8 | 8.8 × 10−10 | 7.8 × 10−11 |
| 3 | 2.2 × 10−3 | 2.5 × 10−5 | 5.0 × 10−7 | 1.4 × 10−8 | 5.1 × 10−10 | 2.2 × 10−11 | 1.1 × 10−12 |
| 4 | 1.3 × 10−3 | 9.3 × 10−6 | 1.2 × 10−7 | 2.1 × 10−9 | 4.6 × 10−11 | 1.2 × 10−12 | 3.9 × 10−14 |
4.2. Partition function of a quantum rotator
4.2.1. Partition function
For a quantum rotator, the eigen energy is
$\begin{eqnarray*}{\epsilon }_{l}=\displaystyle \frac{l(l+1)}{2}{\epsilon }_{1},\qquad l\in {\mathbb{N}}\end{eqnarray*}$
with its degeneracy ωl = 2l + 1, and ε1 is the energy level of the first excited states. The partition function is given by $\begin{eqnarray*}\begin{array}{l}Z=\displaystyle \sum _{l=0}^{+\infty }{\omega }_{l}{{\rm{e}}}^{-\beta {\epsilon }_{l}}=\displaystyle \sum _{l=0}^{+\infty }(2l+1){{\rm{e}}}^{-{ \mathcal B }l(l+1)}\,=\,-\displaystyle \frac{{{\rm{e}}}^{{ \mathcal B }/4}}{{ \mathcal B }}\displaystyle \sum _{l=0}^{+\infty }f^{\prime} (l+\displaystyle \frac{1}{2}),\end{array}\end{eqnarray*}$
with ${ \mathcal B }=\beta {\epsilon }_{1}/2$ and $f(x)={{\rm{e}}}^{-{ \mathcal B }{x}^{2}}$. Compute $\begin{eqnarray}\begin{array}{rcl}Z & = & {Z}_{m}^{\prime} +{{ \mathcal W }}_{{np}}(m+\displaystyle \frac{1}{2})+{R}_{{mnp}},\\ {Z}_{m}^{\prime} & = & \displaystyle \sum _{l=0}^{m-1}(2l+1){{\rm{e}}}^{-{ \mathcal B }l(l+1)},\\ {{ \mathcal W }}_{{np}}(x) & = & {{ \mathcal T }}_{p}(x)+{{ \mathcal U }}_{{np}}(x),\\ {{ \mathcal T }}_{p}(x) & = & \displaystyle \frac{2p+1}{{ \mathcal B }}{{\rm{e}}}^{-{ \mathcal B }({x}^{2}-\tfrac{1}{4})}+2{\left(\displaystyle \frac{\pi }{{ \mathcal B }}\right)}^{3/2}{{\rm{e}}}^{\tfrac{{ \mathcal B }}{4}}\displaystyle \sum _{k=1}^{p}{\left(-1\right)}^{k}k{{\rm{e}}}^{-\tfrac{{\pi }^{2}{k}^{2}}{{ \mathcal B }}}{\rm{Im}}\left(\mathrm{erf}\left(\sqrt{{ \mathcal B }}x-i\displaystyle \frac{k\pi }{\sqrt{{ \mathcal B }}}\right)\right),\\ {{ \mathcal U }}_{{np}}(x) & = & \left(x+\displaystyle \sum _{r=1}^{\rlap{-}n}{\left(-1\right)}^{r+1}{T}_{2r,p}{{ \mathcal B }}^{r-1}{H}_{2r}(\sqrt{{ \mathcal B }}x)\right){{\rm{e}}}^{-{ \mathcal B }({x}^{2}-\tfrac{1}{4})},\end{array}\end{eqnarray}$
where $\mathrm{erf}(x)=\tfrac{2}{\sqrt{\pi }}{\int }_{0}^{x}{{\rm{e}}}^{-{z}^{2}}{\rm{d}}z$ is the error function.4.2.2. Remainder estimation
If equation (29 ) holds, then the remainder can be controlled through equation (30 ) according to corollary 2 as
$\begin{eqnarray}\left|{R}_{{mnp}}\right|\leqslant {\tilde{R}}_{{mnp}}^{H}={T}_{n,p}\left[\displaystyle \frac{n+3}{2}\right]2{G}_{n}{{\rm{e}}}^{-{ \mathcal B }\left(\tfrac{m(m+1)}{2}-\tfrac{1}{8}\right)}{{ \mathcal B }}^{\tfrac{n}{2}-1}\end{eqnarray}$
for n > 1. Again, one can achieve given accuracy by choosing proper m, n, and p.4.3. Partition function of a two-dimensional infinite square well
4.3.1. Partition function
We assume that a particle is in a two-dimensional infinite square well. Then the eigen energies are given by3 to the above summation with $P={{{\mathbb{N}}}^{+}}^{2}$, P0 = {(i, j) ∈ P∣i ≤ m, j ≤ m}, P1 = P − P0, $n^{\prime} =n$, and $p^{\prime} =p$, we obtain26 ) and (27 ), see figure 3.
$\begin{eqnarray*}{\epsilon }_{i,j}={\epsilon }_{\mathrm{1,0}}({i}^{2}+{j}^{2}),\qquad i,j=1,2,3\ldots \end{eqnarray*}$
The partition function is $\begin{eqnarray*}Z=\displaystyle \sum _{i=1}^{+\infty }\displaystyle \sum _{j=1}^{+\infty }{{\rm{e}}}^{-\beta {\epsilon }_{i,j}}=\displaystyle \sum _{i=1}^{+\infty }\displaystyle \sum _{j=1}^{+\infty }{{\rm{e}}}^{-B({i}^{2}+{j}^{2})},\end{eqnarray*}$
with B = βε1,0. Applying theorem $\begin{eqnarray}\begin{array}{rcl}Z & = & \displaystyle \sum _{(i,j)\in {P}_{0}}{{\rm{e}}}^{-B({i}^{2}+{j}^{2})}+\displaystyle \sum _{(i,j)\in {P}_{1}}{{\rm{e}}}^{-B({i}^{2}+{j}^{2})}\\ & = & {Z}_{m}^{2}+{W}_{{np}}(m+1)\left(2{W}_{{np}}(1)-{W}_{{np}}(m+1)\right)+{R}_{{np},{np}},\end{array}\end{eqnarray}$
where Zm and Wnp(x) are defined in equations (4.3.2. Remainder estimation
Now we try to give an upper bound of the remainder with the help of Conjecture 1 . With the help of equation (30 ), we have3 , the remainder can be controlled by37 ), (38 ) as
$\begin{eqnarray}\left|{f}_{{n}_{1},{n}_{2}}(x,y)\right|\leqslant {B}^{\tfrac{{n}_{1}+{n}_{2}}{2}}{G}_{{n}_{1}}{G}_{{n}_{2}}{{\rm{e}}}^{-B\tfrac{{x}^{2}+{y}^{2}}{2}},\end{eqnarray}$
$\begin{eqnarray}\left|{f}_{{n}_{1},0}(x,y)\right|\leqslant {B}^{\tfrac{{n}_{1}}{2}}{G}_{{n}_{1}}{{\rm{e}}}^{-B(\tfrac{{x}^{2}}{2}+{y}^{2})},\end{eqnarray}$
for $f(x,y)={{\rm{e}}}^{-B({x}^{2}+{y}^{2})}$ and ${n}_{1},{n}_{2}\in {{\mathbb{N}}}^{+}$. Note that f(x, y) = f(y, x). According to corollary $\begin{eqnarray*}\begin{array}{rcl}\left|{R}_{{np},{np}}\right|\leqslant {R}_{{np},{np}}^{A} & = & {T}_{n,p}\left(2(2p+1){\displaystyle \int }_{D}\left|{f}_{n,0}(x,y)\right|{\rm{d}}x{\rm{d}}y+{\displaystyle \int }_{\partial D}\left|{f}_{n,0}(x,y){\rm{d}}x\right|\right)\\ & & +2{T}_{n,p}\left(\displaystyle \sum _{r=1}^{n-}{T}_{2r,p}{\displaystyle \int }_{\partial D}\left|{f}_{n,2r-1}(x,y){\rm{d}}x\right|\right)+{T}_{n,p}^{2}{\displaystyle \int }_{D}\ \left|{f}_{n,n}(x,y)\right|{\rm{d}}x{\rm{d}}y.\end{array}\end{eqnarray*}$
The four integrals above can be estimated by equations ( $\begin{eqnarray*}\begin{array}{rcl}{\displaystyle \int }_{D}\left|{f}_{n,0}(x,y)\right|{\rm{d}}x{\rm{d}}y & \leqslant & {B}^{\tfrac{n}{2}}{G}_{n}\xi (\sqrt{B/2},\sqrt{B},m+1),\\ {\displaystyle \int }_{D}\ \left|{f}_{n,n}(x,y)\right|{\rm{d}}x{\rm{d}}y & \leqslant & {B}^{n}{G}_{n}^{2}\xi (\sqrt{B/2},\sqrt{B/2},m+1),\\ {\displaystyle \int }_{\partial D}\left|{f}_{n,0}(x,y){\rm{d}}x\right| & \leqslant & {B}^{\tfrac{n}{2}}{G}_{n}\eta (\sqrt{B/2},\sqrt{B},m+1),\\ {\displaystyle \int }_{\partial D}\left|{f}_{n,2r-1}(x,y){\rm{d}}x\right| & \leqslant & {B}^{\tfrac{n-1}{2}+r}{G}_{n}{G}_{2r-1}\eta (\sqrt{B/2},\sqrt{B/2},m+1),\end{array}\end{eqnarray*}$
with $\begin{eqnarray*}\begin{array}{rcl}\xi (x,y,\nu ) & \equiv & \frac{\pi }{4{xy}}\,\left[\mathrm{erfc}(x)\mathrm{erfc}(\nu y)+\mathrm{erfc}(y)\mathrm{erfc}(\nu x)-\mathrm{erfc}(\nu x)\mathrm{erfc}(\nu y)\right],\\ & & \\ \eta (x,y,\nu ) & \equiv & \frac{\sqrt{\pi }}{2x}\,\left[\mathrm{erfc}(x){{\rm{e}}}^{-{\left(\nu y\right)}^{2}}+\mathrm{erfc}\left(\nu x\right){{\rm{e}}}^{-{y}^{2}}-\mathrm{erfc}(\nu x){{\rm{e}}}^{-{\left(\nu y\right)}^{2}}\right].\\ & & \end{array}\end{eqnarray*}$
Therefore we have $\begin{eqnarray*}\begin{array}{rcl}\left|{R}_{{np},{np}}\right| & \leqslant & {\tilde{R}}_{{np},{np}}^{H}\\ & \equiv & {T}_{n,p}{B}^{\tfrac{n}{2}}{G}_{n}\left(2(2p+1)\xi (\sqrt{B/2},\sqrt{B},m+1)+\eta (\sqrt{B/2},\sqrt{B},m+1)\right.\\ & & \left.+2\displaystyle \sum _{r=1}^{\rlap{-}n}{T}_{2r,p}{B}^{r-\tfrac{1}{2}}{G}_{2r-1}\eta (\sqrt{B/2},\sqrt{B/2},m+1)+{T}_{n,p}{B}^{\tfrac{n}{2}}{G}_{n}\xi (\sqrt{B/2},\sqrt{B/2},m+1\right),\end{array}\end{eqnarray*}$
where n > 1, p ≥ 0, and m ≥ 0. One can achieve any given accuracy by choosing m, n, p properly. Some values of ${\tilde{R}}_{{np},{np}}^{H}$ are listed in table 4.Table 4. ${\tilde{R}}_{{np},{np}}^{H}$ for square well with B = 1 and m = 0. |
| n | 3 | 5 | 7 | 9 | 11 | 13 | 15 |
|---|---|---|---|---|---|---|---|
| p | |||||||
| 0 | 2.2 × 10−2 | 4.7 × 10−3 | 1.6 × 10−3 | 7.2 × 10−4 | 4.0 × 10−4 | 2.6 × 10−4 | 2.0 × 10−4 |
| 1 | 5.8 × 10−3 | 2.6 × 10−4 | 2.1 × 10−5 | 2.3 × 10−6 | 3.1 × 10−7 | 5.1 × 10−8 | 9.6 × 10−9 |
| 2 | 3.2 × 10−3 | 5.8 × 10−5 | 1.9 × 10−6 | 9.0 × 10−8 | 5.3 × 10−9 | 3.8 × 10−10 | 3.2 × 10−11 |
| 3 | 2.1 × 10−3 | 2.1 × 10−5 | 3.7 × 10−7 | 9.5 × 10−9 | 3.1 × 10−10 | 1.2 × 10−11 | 5.7 × 10−13 |
| 4 | 1.6 × 10−3 | 9.7 × 10−6 | 1.1 × 10−7 | 1.7 × 10−9 | 3.5 × 10−11 | 8.7 × 10−13 | 2.6 × 10−14 |
5. Conclusion
In this paper, a MEMF is introduced by working out part of the Fourier expansion of the remainder term in the original Euler–Maclaurin formula, which is also generalized to 2D. The MEMF includes both the original Euler–Maclaurin formula and the Poisson summation formula as special cases, and the remainder term of the MEMF can well be under control when the parameters m, n, p are properly chosen, so that it can be used as alternative way to work out a summation when the integral in the formula can be calculated. Approximate expressions of the partition functions of square well in 1D and 2D, and that of a quantum rotator are obtained with the help of the MEMF.