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A collection of integrals that are deceptively simple to state yet require ingenuity to solve — spanning contour integration, special functions, symmetry arguments, and beyond. Closed forms are given; proofs are left as exercises (or future posts).
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Because the single-variable function \(e^{-x^2}\) lacks an elementary antiderivative, evaluating the Gaussian integral over the real line requires a classic mathematical maneuver: changing the dimension to change the perspective. By evaluating its square in a two-dimensional Cartesian system, we can shift our perspective to radial symmetry.
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By: Rajib Belbase
For students in Calculus II. This post covers the essential ideas behind infinite sequences and series — what they mean, how to test for convergence, and how to represent functions as power series. All material follows Chapter 11 of Calculus, 9th edition, by James Stewart.
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By: Rajib Belbase
For students in Calculus III. This post explores Green’s Theorem, one of the cornerstones of vector calculus. It establishes a profound link between a line integral around a closed curve and a double integral over the region it encloses. All material follows Chapter 16 of Calculus, 9th edition, by James Stewart.
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By: Rajib Belbase
For students in Calculus III. This post covers the essential ideas behind double and triple integrals — what they mean geometrically, how to set them up, and how to compute them. All material follows Chapter 15 of Calculus, 9th edition, by James Stewart.
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By: Rajib Belbase
For students in Calculus I. This post covers the foundational concepts of functions required for Calculus. We explore how to represent functions, determine domains and ranges, and apply transformations. All material follows Sections 1.1–1.3 of Calculus, 9th edition, by James Stewart.
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By: Rajib Belbase
Abstract. Euler’s Identity \(e^{i\pi} + 1 = 0\) is frequently celebrated as the most beautiful equation in mathematics. But its beauty is not mystical. This article dismantles the mysticism and replaces it with something far more satisfying: a rigorous geometric explanation grounded in complex analysis, Taylor series, and the structure of the complex plane. We build every prerequisite from scratch, prove Euler’s formula in full, and examine why the identity is not a coincidence but an inevitability.
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By: Rajib Belbase
Graduate school in the United States is a transformative experience filled with academic rigor, new opportunities, and personal growth. However, it also comes with its fair share of challenges, from managing coursework and research to adjusting to a new cultural and social environment. Here’s a glimpse into the life of a graduate student in the U.S.