Special Functions

Advanced Set Theory

In Advanced Set Theory, we study many important ideas related to the size and structure of sets. We begin by discussing equivalent sets and understanding the difference between countable and uncountable sets, along with their examples. We explore cardinal numbers as a way to compare set sizes and learn about Cantor’s theorem which shows that some infinities are bigger than others. We also look at partially ordered sets, chains, and lattices, which help us understand the order and arrangement of elements. The Cantor-Bernstein theorem helps in proving when two sets have the same size. We learn how to add, multiply, and take powers of cardinal numbers. Important logical tools like the Axiom of Choice and Zorn’s Lemma are also studied, along with their equivalence. Finally, we discuss well-ordered sets, ordinal numbers, and how we can perform addition and multiplication with them. All these concepts help us deeply understand infinite sets and the foundations of modern mathematics. Notes ●...

Special Functions

Special functions are a class of mathematical functions that arise frequently in various fields of science, engineering, and applied mathematics. These functions, such as Bessel functions, Legendre polynomials, Gamma functions, and Hypergeometric functions, often appear as solutions to differential equations that model physical phenomena like heat conduction, wave propagation, and quantum mechanics. Special functions have well-defined properties, including recurrence relations, orthogonality, and integral representations, which make them extremely useful for analytical and numerical problem-solving. Their study helps in simplifying complex expressions and plays a crucial role in mathematical modeling and theoretical research.

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📘 Special Functions (Notes)
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📗 Special Functions by Muzammil Tanveer
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🎥 Special Functions Video Playlist
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