#weierstrass — Public Fediverse posts

Weierstrass Substitution as a Special Case of the USM Framework:
https://geometriadominicana.blogspot.com/2025/12/weierstrass-as-special-case-of-usm.html

#math #calculus #Weierstrass #USM #framework #euler #unification #integral #integration

𝐁𝐢𝐠 𝐔𝐩𝐝𝐚𝐭𝐞 𝐨𝐧 𝐌𝐲 𝐈𝐧𝐭𝐞𝐠𝐫𝐚𝐭𝐢𝐨𝐧 𝐓𝐞𝐜𝐡𝐧𝐢𝐪𝐮𝐞!

I'm excited to share the latest development in what I initially called the "Exponential Substitution Method." Moving forward, this approach will be known as the Unified Substitution Method because it unifies several powerful techniques for tackling integrals, including:

𝟏. 𝐂𝐨𝐦𝐩𝐥𝐞𝐱 𝐄𝐱𝐩𝐨𝐧𝐞𝐧𝐭𝐢𝐚𝐥𝐬 𝐟𝐨𝐫 𝐓𝐫𝐢𝐠𝐨𝐧𝐨𝐦𝐞𝐭𝐫𝐢𝐜 𝐅𝐮𝐧𝐜𝐭𝐢𝐨𝐧𝐬: Originally used to simplify integrals by expressing sine and cosine in terms of exponential functions, this technique is extended to handle irrational integrands.

𝟐. 𝐖𝐞𝐢𝐞𝐫𝐬𝐭𝐫𝐚𝐬𝐬 𝐒𝐮𝐛𝐬𝐭𝐢𝐭𝐮𝐭𝐢𝐨𝐧: Traditionally used to transform rational expressions of trigonometric functions into purely rational ones, this is now extended and incorporated into my method to simplify irrational expressions as well.

𝟑. 𝐄𝐮𝐥𝐞𝐫 𝐒𝐮𝐛𝐬𝐭𝐢𝐭𝐮𝐭𝐢𝐨𝐧𝐬: These classic substitutions, designed for certain irrational integrands, are also integrated into the Unified Substitution Method. Interestingly, I’ve found that the extended Weierstrass substitution often simplifies integrals into rational forms nearly identical to those obtained through Euler’s substitutions.

By unifying and extending these techniques, this method becomes a comprehensive tool capable of handling a wide variety of integrals, including some of the most challenging irrational ones.

This approach represents a significant step forward in simplifying integral calculus and demonstrates the deep interconnections between these classic methods. I’m excited to see how this can benefit students, researchers, and enthusiasts alike.

To stay tuned for more examples and demonstrations of this method, here is my blog post on it:

https://geometriadominicana.blogspot.com/2024/03/integration-using-some-euler-like.html?m=1

#math #calculus #integration #euler #weierstrass #new #method

Today I gave my third talk in the VIG (Very Informal Group) series about Shioda's paper on #Weierstrass Transformations and #Cubic #Surfaces (1994). There are some lovely results in this paper.

@apocheir I remember trying to find the actual letter the whoever "Madame Schabelskoy" was, but I was unsuccessful. I am pretty sure it's originally due to #Weierstrass and #Kovalevskaya paraphrased it.