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Questions tagged [discrete-calculus]

Discrete calculus is an analog of the continuous version where the 'shift parameter' $h$ remains a non-zero positive number instead of being passed to a limit.

Discrete calculus is an analog of real calculus where the difference $h$ does not pass to a limit. For example, compare the usual derivative $f'$ and the discrete (forward-) difference $\Delta$: $$ f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}\qquad \Delta f(x):= f(x+1)-f(x) $$Many continuous objects have a discrete analog. For instance, $\Delta 2^n = 2^n$, and the Bernoulli polynomials $B_n(x)$ satisfy $\Delta B_n(x) = n x^{n-1}$. Definite and indefinite integrals can be defined as well using infinite series.

Discrete calculus is used in pure and applied areas, ranging from image processing to number theory. Consider using other tags to hone the focus of your question.

163 questions
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Is there a discrete version of de l'Hôpital's rule?

When considering asymptotics of runtime functions, you often have to find limits of quotients of discrete functions, e.g. $\displaystyle\qquad \lim\limits_{n \to \infty} \frac{4^n}{\binom{2n}{n}\sqrt{n}}.$ While this particular case can easily be…
Raphael
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What are the $\sin$ and $\cos$ of discrete calculus?

I'm getting acquainted with Discrete Calculus, and I really like taking functions that arise in traditional calculus and finding what their counterparts in discrete-land are. For example, if we define our difference operator $$\Delta f(n) = f(n + 1)…
Eli Rose
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Changing Variables in Discrete Calculus

In discrete calculus one soon meets the $h$-difference operator $$\Delta_h[f(x)] = f(x+h) - f(x)$$ and we often define $\Delta = \Delta_1.$ We can similarly define the indefinite sums $\Delta_h^{-1}$ and set $\Delta^{-1} = \Delta_1^{-1}.$ Most books…
Kariuki
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Is there a non-constant function $f$ such that $f'(x) = f(x - 1)$?

In discrete calculus, where the difference operator $\Delta f = f(x + 1) - f(x)$ takes the place of $\frac{d}{dx}$, Fibonacci sequences are given by the functions satisfying: $$ \Delta f(x) = f(x - 1) $$ Is there a non-constant function such that…
Eli Rose
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Purely "discrete" PDEs?

Usually, one formulates a system of continuous PDEs and then discretizes it in order to approximately solve it. Is there a view point that instead formulates a system of "discrete" PDEs, which therefore do not require a discretization step in order…
bzm3r
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Using discrete calculus to study convergence of series and sequences

From some personal investigation, I've noticed that all convergence tests for infinite series (at least, the real kind) can be rephrased in terms of the discrete derivative $∆f(x)$ of a function $f(x)$, sometimes to give interesting results. For…
GregRos
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Comparing discrete sums and integrals

Integral is the summation of continuous numbers. Taking for example the graph for $y=x$ for $x$ between $[0,2]$, the area/integral is $2$. Now if I take summation for discrete $y=x$, $x \in \mathbb{Z}$ between $[0,2]$, the summation is 3. What I'm…
akipro
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Limit of a function with discrete codomain

Let $f(n) = \sum_{i=0}^n 2^{-i}$ Let $g(n) = \begin{cases} \text{False}, & f(n) < 2\\ \text{True}, & f(n) \geq 2 \end{cases}$ I’m trying to find $\lim_{n \to \infty} g(n)$ Clearly $\lim_{n \to \infty} f(n) =2$ I do not think it is valid to say that…
Hugh
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Discretization of an integral

Given $f: [a,b] \to R $ and $K: [a,b]$ x $[a,b]$ $\to R$, we want to find a solution $\varphi:[a,b] \to R $ to the Fredholm integral equation: $$\varphi(x) = f(x)+\int _{ a }^{ b }{ K( x,t)\varphi (t)dt } $$ For a given integer n, we will consider…
Joe
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Discrete calculus: is my proof about the difference of two consecutive powers correct?

Theorem statement: $\displaystyle \Delta^dF_n(x) = \sum_{k=0}^{n-1} \binom{n-1}{k} \Delta^{d-1}F_{n-1-k}(x)(-1)^k$ Proof: $$\begin{align} F_n(x) &= x^n =\Delta^0F(x) \\[2ex] \Delta^1F_n(x) &= x^n -(x-1)^n \\[2ex] &= x^n - \sum_{k=0}^n\binom nk…
user110391
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How to get $(n+1)^{\underline k} - n^{\underline k} = n^{\underline{k-1}}$?

How I've understood it, a falling factorial, $n^{\underline k} = \frac{n!}{k!} = n(n-1)(n-2)...(n-k+1) = \displaystyle\prod_{i=k+1}^n i$ So, $$\Delta_n (n^{\underline k}) = (n+1)^{\underline k} - n^{\underline k} = \frac{(n+1)!}{k!} -…
user110391
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Is there something called the pochhammer expansion?

I'm just playing around with the Pochhammer notation where $$n^{\underline{k}} \equiv n(n-1)\ldots(n-k+1).$$ I've established the formulas $$\begin{split} n &= n^{\underline{1}}\\ n^2 &= n^{\underline{2}} + n^{\underline{1}}\\ n^3 &=…
9cco
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discrete Laplacian on nonuniform rectangle grid

I wonder if there is a way to extend the finite difference discretization of the Laplacian on a uniform grid to a nonuniform grid. More specifically, I am not sure that the finite differences approach is the right approach for my problem, where I…
Yohai Magan
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How to discretize a certain integral

I would like to discretize the following integral operator: $$\frac{1}{s^2}\sum_{j=1}^N\mu_j\int d\mathbf{x}d\mathbf{x}'f(\mathbf{x})f(\mathbf{x'}) \left(x_j + x'_j -…
Sid
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Approximation of "smooth" discrete functions

Assume $f : \{1,\ldots,n\} \to \mathbb{C}$ satisfies $ |D^\ell f(i)| \leq C $ for all $i \in \{1,\ldots,n-\ell\}$ and all $\ell \in \{0, \ldots, k\}$ for some $k \in \mathbb{N}$. Here, $D$ denotes the finite difference operator $$ Df(i) := f(i+1)…
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