miércoles, 17 de julio de 2024

Showing $\frac{1}{e^{i\alpha_2}} + \frac{1}{e^{i\beta_2}} + \frac{1}{e^{i\gamma_2}} = \frac{1}{e^{i(\alpha_2+\beta_2+\gamma_2)}}$

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Let $x$ be any of $\alpha_1$, $\beta_1$, or $\gamma_1$ and suppose $\alpha_1+\beta_1+\gamma_1=\pi$. Then $$e^{i(\alpha_2+\beta_2)}+e^{i(\alp...
viernes, 5 de julio de 2024

Integrals of the form $\int f\left(x,\frac{\sqrt{x+m}}{\sqrt{x+n}}\right)\,dx$

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We give a general approach for integrals of the form $\int f\left(x,\frac{\sqrt{x+m}}{\sqrt{x+n}}\right)\,dx$. Let $b-a=m$ and $b+a=n$, whe...
jueves, 21 de marzo de 2024

A new integration technique via Euler-like identities

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"Complexification formulas are great and it seems like this simplifies the right away." - Ninad Munshi We introduce Exponential Su...
miércoles, 28 de febrero de 2024

A family of trigonometric formulas for the roots of quadratic equations

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  This note presents alternative trigonometric formulas for finding the roots of quadratic equations where $a$, $b$, and $c$ are non-zero re...
jueves, 15 de febrero de 2024

Integrals yielding $e^{\pi}$ or $e^{-\pi}$

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 Lately, I've been playing a lot with integrals , and coincidentally (with a bit of algebraic manipulation), I've come across these ...
miércoles, 7 de febrero de 2024

Solving 'impossible' integrals with a new trick

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"Complexification formulas are great and it seems like this simplifies the right away." Ninad Munshi The following identities have...
sábado, 13 de enero de 2024

Trigonometric formula for solving quadratic equations

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In this note, we provide an alternative trigonometric formula for solving quadratic equations where $a$, $b$, and $c$ are non-zero real num...