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Theory of Equations Introduction In this blog, we will dive into the Theory of Equations, with a focus on polynomial equations, which are essential in both mathematics and real-world applications. The Theory of Equations plays a crucial role in understanding how to find the roots (solutions) of an equation and explore the relationship between these […]
Legendre Polynomial Show that the coefficient of ( x^n ) in ( P_n(x) ) is ( frac{(2n)!}{2^n(n!)^2} ) Solution: The n-th Legendre Polynomial is given by [ P_n(x)=frac{1}{2^nn!}frac{d^n}{dx^n}left( x^2-1right)^n ] Expanding ( left( x^2-1 right) ) using Binomial theorem, we get begin{align*} P_n(x) &=frac{1}{2^nn!}frac{d^n}{dx^n}left[ x^{2n}-nx^{2(n-1)}+cdots right]\[0.2cm] P_n(x) &=frac{1}{2^nn!}left[ frac{d^n}{dx^n} left( x^{2n}right) -cdots right]\[0.2cm] P_n(x) &=frac{1}{2^nn!}left[
Taylor Series and Maclaurin Series Explained: Derivations of Maclaurin Series of 10 Essential Functions Table of Contents Introduction Calculus, the language of change, equips us with powerful tools to analyze and understand how things move, grow, and interact. Imagine a scenario where you need to determine the future position of a speeding car or
LaTeX Unlocked: A Powerful Blueprint for Beginners to Conquer Document Design! (Part 2) Table of Contents Introduction In the first part of this blog, we covered the fundamentals of LaTeX, from installation and basic structure to document formatting and typography. Building on this foundation, the second section of our investigation delves into more advanced
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LaTeX Unlocked: A Powerful Blueprint for Beginners to Conquer Document Design! (Part 1) Table of Contents Introduction Welcome to the world of LaTeX, a sophisticated typesetting system that turns your writings into polished, professional works of art. LaTeX provides a versatile and fast way to write documents with precision and flair, whether you’re a
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From Fundamentals to Advanced: 50+ Integration Formulas Explored Your Web Page Table of Contents Introduction Unlock the power of mathematical analysis with Integration Formulas, a crucial element in calculus. Integration is a mathematical operation that involves finding the integral of a function, representing the area under its curve on a given interval. It is the
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Integration of ln x in 5 simple steps using Integration by Parts Step 1: Start with the integral of (ln(x)) with respect to (x): [ int ln(x) , dx ] Step 2: Use integration by parts, which states: [ int u , dv = uv – int v , du ] In this case, we’ll
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