China
Africa
Explore chapters and articles related to this topic
Properties of ℝ
Published in John D. Ross, Kendall C. Richards, Introductory Analysis, 2020
John D. Ross, Kendall C. Richards
A primary goal of a first course in real analysis is to develop the theory of calculus with a rigorous mathematical approach. The “rigorous mathematical” approach is key here: oftentimes in a standard first-year calculus course, students learn rules and techniques without developing why these rules work. One of our fundamental goals is to guide the student in developing proofs of these rules. We will do this by proving mathematical statements which assert that a conclusion of interest is true when a given hypothesis is satisfied. Such statements are usually called theorems. We sometimes replace the word theorem with corollary (meaning “theorem that follows quickly from previous work) or lemma (meaning “mini theorem” used to prove a result of primary interest). To state our theorems with clarity, we will need definitions (i.e., statements that introduce the concise terminology used to describe important mathematical objects and properties). We will also need a starting point, namely collection of axioms which are mathematical statements that we accept as true, without proof. Using this language, the goal of this text (and the goal of a beginning real analysis student) is this:
Complex Analysis
Published in Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene, Modern Engineering Mathematics, 2017
Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene
As we know the real analysis or calculus is the study of functions of real variables (f : R → R or f : R2 → R). Similarly complex analysis is the study of functions of complex variables f : C → C, where C denotes the set of complex numbers z = x + iy, x ∈ R, y ∈ R) and the imaginary number i=−1 is the root of the algebraic equation x2 + 1 = 0. It is well known that complex analysis was developed as a result of mathematical curiosity but subsequently it was found very useful in signal and image processing, fluid flow, quantum mechanics and many other areas of engineering.
On studying equivalent (or not) definitions; the case of limits in R and R 2
Published in International Journal of Mathematical Education in Science and Technology, 2022
The paper underlines the significance of building up alternative definitions of fundamental mathematical concepts and reconciling known ones that are not equivalent. The mathematical environment is the one of Real Analysis; it focuses on the concept of limit in and . The instructional medium is a series of tasks assigned to students to work in a class directed by Analysis lecturers. The justification of this pedagogical outlook is grounded on mathematical and cognitive concerns, as well as on the usability of the resulting definitional statements in mathematical work. We hope that our proposal will be acknowledged as constructive and beneficial for attaining concepts at an advanced level in Mathematics.