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Bessel’s function

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The detailed proofs of the key properties of Bessel functions of the first kind \( J_n(x) \) and the second kind \( Y_n(x) \). We’ll focus on:ODE derivationSeries expansion derivation for \( J_n(x) \)Parity proofRecurrence relationsDifferentiation identitiesAsymptotic behaviorDefinition and properties of \( Y_n(x) \) Bessel’s Differential EquationWe start from the ODE:\[x^2 y” + x y’ … Read more

Visualize Waveguide Power Distribution: An Interactive Web Tool

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Electromagnetic waves confined within waveguides are fundamental to microwave engineering, telecommunications, and particle accelerators, yet visualizing the invisible fields and understanding how different modes combine within the guide can be challenging. To bridge this gap, we introduce the Waveguide Pattern Visualizer, an interactive web application designed specifically to simulate and display the power distribution across … Read more

Converting WEBM to MP3: Why and How

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Why Convert WEBM to MP3? The .webm format is a popular choice for storing audio and video due to its high compression and support for modern codecs. However, .mp3 remains the most widely supported and versatile audio format, compatible with virtually all devices, media players, and online platforms. Here are some reasons why you might … Read more

Finding the Symmetric Point of a Point About a Line

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In geometry, the concept of symmetry about a line is a fascinating topic with applications in various fields. This article presents two methods—coordinate geometry and vector algebra—to find the symmetric point \( Q \) of a point \( P(m, n) \) about the line through points \( A(x_0, y_0) \) and \( B(x_1, y_1) \). … Read more

How to Render AsciiMath and LaTeX in HTML Pages?

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Displaying complex mathematical equations, such as integrals, summations, and matrices, directly in HTML pages is possible with the help of JavaScript libraries like KaTeX or MathJax. KaTeX is a JavaScript library specifically designed to render mathematical expressions in web browsers. It’s lightweight, fast, and provides high-quality rendering of LaTeX math. This makes it an ideal … Read more

how to use jQuery in WordPress

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WordPress already includes jQuery by default. However, it follows a slightly different approach to ensure compatibility with other JavaScript libraries. Here’s how you can use jQuery in WordPress: 1. Enqueue jQuery Properly First, make sure you’re enqueuing jQuery in your theme or plugin. WordPress includes its own version of jQuery, so you should load it … Read more

Understanding Coordinate Systems in Mathematics and Programming Language

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In traditional mathematics, the Cartesian coordinate system is commonly used and is defined as follows: This system is intuitive for plotting functions, geometric shapes, and understanding spatial relationships. It’s the standard used in mathematics textbooks and many scientific contexts. Coordinate Systems in Programming In computer graphics and many programming environments (including p5.js, HTML5 Canvas, and … Read more

The mathematics of the smith chart

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The Smith chart is a graphical tool used in electrical engineering, particularly in RF (radio frequency) and microwave engineering, to represent complex impedance(~Z~), admittance(~Y~), and reflection coefficients(~\Gamma~). It provides a way to visualize and solve problems involving transmission lines and matching circuits without requiring extensive calculations. Key Aspects of the Smith Chart: Applications: By using … Read more