Sequence Curves: Essential Properties of Essence

In my last post I introduced my goals of $Ȣf(x)$, a transformation that produces a new function that fits the definition $\sum_{i=n}^mf(i) = \int_{n-1}^mȢf(x)dx$. In this article, we will explore few basic properties we can find of Ȣ Transformation without even needing to know how to transform works!

Sum Rule of Essence 

$$Ȣ(f(x)+g(x)) = Ȣf(x)+Ȣg(x)$$ This property becomes very obvious when we consider that Essence is related to Series.

Let's consider $Ȣ(f(x)+g(x))$. By definition of Essence, we find that
$$\sum_{i=n}^m(f(i)+g(i)) = \int_{n-1}^mȢ(f(x)+g(x))dx$$
We can modify the Summation on the left-hand side as such
$$\sum_{i=n}^m(f(i)+g(i)) = \sum_{i=n}^mf(i) + \sum_{i=n}^mg(i)$$
Using Essence, we can represent each sub-summations as integrals
$$\begin{align*}\sum_{i=n}^m(f(i)+g(i)) = & \sum_{i=n}^mf(i) + \sum_{i=n}^mg(i)\\ = & \int_{n-1}^mȢf(x)dx+\int_{n-1}^mȢg(x)dx\\ = & \int_{n-1}^mȢf(x)+Ȣg(x)dx \end{align*}$$
This leads to that $$\sum_{i=n}^m(f(i)+g(i)) = \int_{n-1}^mȢ(f(x)+g(x))dx = \int_{n-1}^mȢf(x)+Ȣg(x)dx\\ \therefore Ȣ(f(x)+g(x)) = Ȣf(x)+Ȣg(x) \\ \blacksquare$$

Constant Multiple Rule of Essence

$$Ȣ(c*f(x)) = c*Ȣf(x), \text{ $c$ is a constant}$$

Let's start with the left hand side, utilizing definition of Essence as
$$\sum_{i=n}^mc*f(i) = \int_{n-1}^mȢ(c*f(x))dx$$
We can modify the Summation such that it leads to
$$\begin{align*} \sum_{i=n}^mc*f(i) = &c*\sum_{i=n}^mf(i) \\ =& c*\int_{n-1}^mȢf(x)dx\\ =& \int_{n-1}^mc*Ȣf(x)dx \end{align*}$$
This leads to
$$\sum_{i=n}^mc*f(i) = \int_{n-1}^mȢ(c*f(x))dx = \int_{n-1}^mc*Ȣf(x)dx \\ \therefore Ȣ(c*f(x)) = c*Ȣf(x) \\ \blacksquare$$

Index Shifting of Essence 

$$Ȣf(x+c) = Ȣ(y), y = x+c$$ This is to say that given $$Ȣf(x+c) = g(x)$$ we can rightly say that $$Ȣf(x) = g(x-c)$$ This is obvious and may seem unnecessary to include in this list, but the ability to utilize this 'shifting' will become extremely useful later on.

Derivative and Integral Rule of Essence

After we cover in more depth a Formal Definition of Essence Transformation, we will be able to cover more unintuitive properties such as
$$\frac{d}{dx}Ȣf(x) = Ȣ\frac{d}{dx}f(x)\\ \intȢf(x)dx = Ȣ\int f(x)dx$$
We will come back to proofs of these after we cover Formal Definition of Essence.


This leaves us with this List of Properties
$$\begin{align*} &Ȣc*f(x) = c*Ȣf(x), \text{$c$ is a constant}\\ &Ȣf(x) + Ȣg(x) = Ȣ(f(x) + g(x))\\ &Ȣf(x+c) = Ȣf(y), y = x+c \\ &\frac{d}{dx}Ȣf(x) = Ȣ\frac{d}{dx}f(x)\\ &\int Ȣf(x) dx = Ȣ\int f(x) dx \end{align*}$$

Being able to utilize these properties freely will help us understand and explore Sequence Curve with more ease!