Properties of Summations and Their Applications in Discrete Calculus
Abstract
Working with sums of numbers is common in many problems faced daily by specialists in various branches of knowledge. Determining these sums involves obtaining compact expressions from a theoretical standpoint. Software applications offer facilities to avoid data capture errors.
This work compiles summation properties reported in the literature, considering their broad applications. It proposes and demonstrates another set particularly relevant when working with discrete variable functions with uniform variation ranges across their domain.
I. Introduction
Studying phenomena and processes in nature and society leads to models describing and predicting their behavior. These models can be grouped into two categories: continuous (e.g., describing wave movement, vehicle displacement) or discrete (e.g., milestone payments, temperature records).
The discrete category is significant today due to the rapid development of digital technology, where information is represented by ordered sets of two logical values: false or true.
Mathematically, studying functions with discrete dependent variable changes involves summations and sets.
This paper presents a set of properties from the literature on summations and derives others that can simplify calculations, such as solving systems of linear equations for obtaining analytical expressions for discrete independent variable function derivatives.
II. General
Summation refers to the sum of a finite set of numbers, denoted as follows:
$$S = \sum_{k=h}^{h+t} n_k$$
where:
- S: Resulting sum.
- T: Number of values to add.
- k: Summation index, varying between h and h + t.
- h: Initial summation point.
- h + t: End point of the summation.
- nk: Value of the object of magnitude at point k.
A particular type of summation of great importance is when t → ∞, known as a series and represented as follows:
$$S = \sum_{k=h}^{\infty} n_k$$
Considering the extensive nature of series analysis, this work does not address this topic.
III. Properties of Summations
Among the general properties of summations reported in the literature, eleven are listed below, proven using mathematical induction.
III.1 Reported in the Literature
Property # 1:
**(Insert mathematical expression for Property #1 here)**
Property # 2:
**(Insert mathematical expression for Property #2 here)**
Property # 3:
**(Insert mathematical expression for Property #3 here)**
Property # 4:
**(Insert mathematical expression for Property #4 here)**
Property # 5:
**(Insert mathematical expression for Property #5 here)**
Property # 6:
**(Insert mathematical expression for Property #6 here)**
Property # 7:
**(Insert mathematical expression for Property #7 here)**
Property # 8:
**(Insert mathematical expression for Property #8 here)**
Property # 9:
**(Insert mathematical expression for Property #9 here)**
Property # 10:
**(Insert mathematical expression for Property #10 here)**
Property # 11:
**(Insert mathematical expression for Property #11 here)**
III.2 Obtained in This Work
Many problems require calculating summations that meet special requirements, such as solving systems of linear equations for determining derivatives of functions with uniform dependent variable variation intervals, problems exhibiting symmetry, etc. Under these conditions, obtaining useful expressions simplifies calculations. Some deduced properties are noted below.
III.2.1 Considering Symmetry in the Path of the Index of Summation
Symmetry simplifies calculations in models of phenomena or processes. In summations, this feature corresponds to the sum index ranging as follows:
$$k = -\frac{t}{2}, -\frac{t}{2} + 1, …, -1, 0, 1, …, \frac{t}{2} – 1, \frac{t}{2}$$
Under this hypothesis, it is possible to obtain the following properties:
Property # 1:
**(Insert mathematical expression for Property #1 here)**
Proof:
**(Insert mathematical proof for Property #1 here)**
Property # 2:
**(Insert mathematical expression for Property #2 here)**
Proof:
**(Insert mathematical proof for Property #2 here)**
Property # 3:
**(Insert mathematical expression for Property #3 here)**
Property # 4:
**(Insert mathematical expression for Property #4 here)**
Property # 5:
**(Insert mathematical expression for Property #5 here)**
III.2.2 Solving Systems of Linear Equations with Independent Variable of the Form x ± kΔx
One application where symmetrical summations are relevant is obtaining analytical expressions for calculating derivatives of discrete variable functions, which commonly involve terms raised to a certain power. The following five properties are derived from practical use.
Property # 1: Calculation of
**(Insert mathematical expression for Property #1 here)**
Property # 2: Calculation of
**(Insert mathematical expression for Property #2 here)**
Property # 3: Calculation of
**(Insert mathematical expression for Property #3 here)**
Property # 4: Calculation of
**(Insert mathematical expression for Property #4 here)**
Property # 5: Calculation of
**(Insert mathematical expression for Property #5 here)**
Property # 6: Calculation of
**(Insert mathematical expression for Property #6 here)**
IV. Conclusions
This work described a set of properties related to summations from the literature, from which various other properties were derived. These are particularly useful for calculating determinants associated with solving linear equation systems resulting from obtaining analytical expressions for calculating discrete variable function derivatives.
V. Bibliography
- Challis, JS, Clarke, GM:”Mathematical Analysis of the Gaussian and Lorentzian Incremental Second Derivative Function”, Spectrochimica Acta, vol 21 pp :791-797, 1965.
- Dixit, L., Ram, S.:”Quantitive Analysis by Derivative Electronics Spectroscopy” Applied Sprectroscopy Reviews, vol 21, # 4, pp :311-418, 1985.
- Faddeev, DK; Faddeva, VN: ‘Computational Methods of Linear Algebra”, Ediciones Revolutionary Cuba, 1971.
- Fraser, RDB, Suzuki, E.:”Resolution of Overlapping Bands: Functions for Simulating Bands Shape”, Analytical Chemistry, vol 41 # 1, pp :37-39, ene/69.
- Glez, MO; Mancilla, JD:”Elementary Modern Algebr”, Educational Publishing, Third Edition, Vol · II, Cuba, 1961
- Mesa, J.; Bermello, A.: ‘Calculation of the derivatives of up to fourth order discrete variable functions,”in preparation to send monografías.com
- Samarskie, AA,”Introduction to Numerical Method”, Editorial MIR, Moscow, 1986.
- Spivak, M.: ‘Calculus’, Publishing Revolutionary Cuba, 1974.
- Suárez, M.:”Numerical Mathematic”, Pueblo y Educación, Cuba 1982.
- Taylor, AE,”Advanced Calculu”, Ediciones Revolutionary Cuba, 1968.
- ASM Handbook of Engineering Mathematics, American Society of Metals, USA, 1983.
Work sent by:
M.Sc. Mr. Jesus Mesa Orama,
Policies and Procedures Specialist, Society Havanatur SA, Corporacion Cimex SA