#halfangleapproach — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #halfangleapproach, aggregated by home.social.
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When a calculus student ends up solving this integral using partial fraction decomposition (PFD) or integration by parts (IBP), what they have actually done is successfully applied a classical algorithm. It’s like solving the Rubik’s cube by following an algorithm you found on the Internet: the heavy lifting of inventing PFD and IBP was done centuries ago by Bernoulli, Leibniz, and Gregory.
That said, the USM is, as far as I know, the only published general scheme that, for this kind of radical–rational integrals, systematically reduces the integrand to polynomial-type functions (Laurent polynomials, to be more precise). There’s no need to set up complicated PFDs or wrestle with sec³, which can be tricky with IBP. That’s what I call solving a problem, if you’ll allow me a bit of self-promotion.
You can find a first draft (I’ll soon upload a second version with benchmarks and some improvements) of the USM method on arXiv: https://arxiv.org/abs/2505.03754
#calculus #math #symmetrymatters #halfangleapproach #euler #integration
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When a calculus student ends up solving this integral using partial fraction decomposition (PFD) or integration by parts (IBP), what they have actually done is successfully applied a classical algorithm. It’s like solving the Rubik’s cube by following an algorithm you found on the Internet: the heavy lifting of inventing PFD and IBP was done centuries ago by Bernoulli, Leibniz, and Gregory.
That said, the USM is, as far as I know, the only published general scheme that, for this kind of radical–rational integrals, systematically reduces the integrand to polynomial-type functions (Laurent polynomials, to be more precise). There’s no need to set up complicated PFDs or wrestle with sec³, which can be tricky with IBP. That’s what I call solving a problem, if you’ll allow me a bit of self-promotion.
You can find a first draft (I’ll soon upload a second version with benchmarks and some improvements) of the USM method on arXiv: https://arxiv.org/abs/2505.03754
#calculus #math #symmetrymatters #halfangleapproach #euler #integration
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When a calculus student ends up solving this integral using partial fraction decomposition (PFD) or integration by parts (IBP), what they have actually done is successfully applied a classical algorithm. It’s like solving the Rubik’s cube by following an algorithm you found on the Internet: the heavy lifting of inventing PFD and IBP was done centuries ago by Bernoulli, Leibniz, and Gregory.
That said, the USM is, as far as I know, the only published general scheme that, for this kind of radical–rational integrals, systematically reduces the integrand to polynomial-type functions (Laurent polynomials, to be more precise). There’s no need to set up complicated PFDs or wrestle with sec³, which can be tricky with IBP. That’s what I call solving a problem, if you’ll allow me a bit of self-promotion.
You can find a first draft (I’ll soon upload a second version with benchmarks and some improvements) of the USM method on arXiv: https://arxiv.org/abs/2505.03754
#calculus #math #symmetrymatters #halfangleapproach #euler #integration
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Hey, I just generalized the 𝐥𝐚𝐰 𝐨𝐟 𝐜𝐨𝐭𝐚𝐧𝐠𝐞𝐧𝐭𝐬! Take a look at it on my blog: https://geometriadominicana.blogspot.com/2025/08/generalization-of-law-of-cotangents.html
#math #geometry #trigonometry #halfangleapproach #symmetrymatters
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Hey, I just generalized the 𝐥𝐚𝐰 𝐨𝐟 𝐜𝐨𝐭𝐚𝐧𝐠𝐞𝐧𝐭𝐬! Take a look at it on my blog: https://geometriadominicana.blogspot.com/2025/08/generalization-of-law-of-cotangents.html
#math #geometry #trigonometry #halfangleapproach #symmetrymatters
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Hey, I just generalized the 𝐥𝐚𝐰 𝐨𝐟 𝐜𝐨𝐭𝐚𝐧𝐠𝐞𝐧𝐭𝐬! Take a look at it on my blog: https://geometriadominicana.blogspot.com/2025/08/generalization-of-law-of-cotangents.html
#math #geometry #trigonometry #halfangleapproach #symmetrymatters
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Hey, I just generalized the 𝐥𝐚𝐰 𝐨𝐟 𝐜𝐨𝐭𝐚𝐧𝐠𝐞𝐧𝐭𝐬! Take a look at it on my blog: https://geometriadominicana.blogspot.com/2025/08/generalization-of-law-of-cotangents.html
#math #geometry #trigonometry #halfangleapproach #symmetrymatters
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Hey, I just generalized the 𝐥𝐚𝐰 𝐨𝐟 𝐜𝐨𝐭𝐚𝐧𝐠𝐞𝐧𝐭𝐬! Take a look at it on my blog: https://geometriadominicana.blogspot.com/2025/08/generalization-of-law-of-cotangents.html
#math #geometry #trigonometry #halfangleapproach #symmetrymatters
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USM really does slash the rote burden, chiefly because one handful of exponential/hyperbolic identities replaces a patchwork of separate trig, inverse-trig, radical and Euler recipes.
Article (draft): https://arxiv.org/abs/2505.03754
#math #calculus #integral #new #euler #arxiv #halfangleapproach #symmetrymatters
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USM really does slash the rote burden, chiefly because one handful of exponential/hyperbolic identities replaces a patchwork of separate trig, inverse-trig, radical and Euler recipes.
Article (draft): https://arxiv.org/abs/2505.03754
#math #calculus #integral #new #euler #arxiv #halfangleapproach #symmetrymatters
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USM really does slash the rote burden, chiefly because one handful of exponential/hyperbolic identities replaces a patchwork of separate trig, inverse-trig, radical and Euler recipes.
Article (draft): https://arxiv.org/abs/2505.03754
#math #calculus #integral #new #euler #arxiv #halfangleapproach #symmetrymatters
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The first and most complete unification of classical integration techniques ever! Say hello to USM.
Draft article: https://drive.google.com/file/d/12DayP6cD1VwDIZCL-nMlcaNH2XUwHfAy/view?usp=drivesdk
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The first and most complete unification of classical integration techniques ever! Say hello to USM.
Draft article: https://drive.google.com/file/d/12DayP6cD1VwDIZCL-nMlcaNH2XUwHfAy/view?usp=drivesdk
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The first and most complete unification of classical integration techniques ever! Say hello to USM.
Draft article: https://drive.google.com/file/d/12DayP6cD1VwDIZCL-nMlcaNH2XUwHfAy/view?usp=drivesdk
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The first and most complete unification of classical integration techniques ever! Say hello to USM.
Draft article: https://drive.google.com/file/d/12DayP6cD1VwDIZCL-nMlcaNH2XUwHfAy/view?usp=drivesdk
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The first and most complete unification of classical integration techniques ever! Say hello to USM.
Draft article: https://drive.google.com/file/d/12DayP6cD1VwDIZCL-nMlcaNH2XUwHfAy/view?usp=drivesdk
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The USM, the Dominican method 🇩🇴, has relegated Euler substitutions to mere historical relics. Modern integration has a new name.
Method draft: https://drive.google.com/file/d/12DayP6cD1VwDIZCL-nMlcaNH2XUwHfAy/view?usp=drivesdk
#math #calculus #integrals #method #euler #halfangleapproach #symmetrymatters -
The USM, the Dominican method 🇩🇴, has relegated Euler substitutions to mere historical relics. Modern integration has a new name.
Method draft: https://drive.google.com/file/d/12DayP6cD1VwDIZCL-nMlcaNH2XUwHfAy/view?usp=drivesdk
#math #calculus #integrals #method #euler #halfangleapproach #symmetrymatters -
The USM, the Dominican method 🇩🇴, has relegated Euler substitutions to mere historical relics. Modern integration has a new name.
Method draft: https://drive.google.com/file/d/12DayP6cD1VwDIZCL-nMlcaNH2XUwHfAy/view?usp=drivesdk
#math #calculus #integrals #method #euler #halfangleapproach #symmetrymatters -
The USM 𝐢𝐧𝐜𝐨𝐫𝐩𝐨𝐫𝐚𝐭𝐞𝐬, 𝐞𝐱𝐭𝐞𝐧𝐝𝐬, 𝐣𝐮𝐬𝐭𝐢𝐟𝐢𝐞𝐬, and 𝐬𝐮𝐫𝐩𝐚𝐬𝐬𝐞𝐬 Euler's substitutions. In Euler's substitutions, the choice of signs based on the domain must be made manually, whereas in the USM, the supporting theorems prescribe which sign to use according to the domain. Moreover, the USM shows that Weierstrass substitutions and the use of complex exponentials for integration are merely two sides of the same coin. The USM not only 𝐮𝐧𝐢𝐟𝐢𝐞𝐬 these two techniques into one, but also 𝐠𝐞𝐧𝐞𝐫𝐚𝐥𝐢𝐳𝐞𝐬 them.
USM: https://geometriadominicana.blogspot.com/2024/03/integration-using-some-euler-like.html
#math #calculus #integrals #technique #new #halfangleapproach #symmetry
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The USM is superior to traditional methods!
Using the USM’s IV transformation formula, from the summary at the link below, you convert integrals of this type into polynomial-type integrals. The traditional method reduces it to the integral of csc³, which would require complicated reduction formulas or integration by parts. The third Euler substitution reduces it to the integral of √(t² + a²), which also requires memorizing a standard formula or integrating it from scratch. None of this is as simple as integrating a three-term polynomial-type function, which is taught in the first few weeks of any integral calculus course.
Check out the technique here: http://geometriadominicana.blogspot.com/2024/03/integration-using-some-euler-like.html
#math #calculus #integration #USM #newmethod #halfangleapproach #symmetrymatters -
The USM is superior to traditional methods!
Using the USM’s IV transformation formula, from the summary at the link below, you convert integrals of this type into polynomial-type integrals. The traditional method reduces it to the integral of csc³, which would require complicated reduction formulas or integration by parts. The third Euler substitution reduces it to the integral of √(t² + a²), which also requires memorizing a standard formula or integrating it from scratch. None of this is as simple as integrating a three-term polynomial-type function, which is taught in the first few weeks of any integral calculus course.
Check out the technique here: http://geometriadominicana.blogspot.com/2024/03/integration-using-some-euler-like.html
#math #calculus #integration #USM #newmethod #halfangleapproach #symmetrymatters -
The USM is superior to traditional methods!
Using the USM’s IV transformation formula, from the summary at the link below, you convert integrals of this type into polynomial-type integrals. The traditional method reduces it to the integral of csc³, which would require complicated reduction formulas or integration by parts. The third Euler substitution reduces it to the integral of √(t² + a²), which also requires memorizing a standard formula or integrating it from scratch. None of this is as simple as integrating a three-term polynomial-type function, which is taught in the first few weeks of any integral calculus course.
Check out the technique here: http://geometriadominicana.blogspot.com/2024/03/integration-using-some-euler-like.html
#math #calculus #integration #USM #newmethod #halfangleapproach #symmetrymatters -
𝗧𝗵𝗲 𝗳𝗶𝗿𝘀𝘁 𝗶𝗻𝘁𝗲𝗴𝗿𝗮𝘁𝗶𝗼𝗻 𝘁𝗲𝗰𝗵𝗻𝗶𝗾𝘂𝗲 𝗱𝗶𝘀𝗰𝗼𝘃𝗲𝗿𝗲𝗱 𝗶𝗻 𝘁𝗵𝗲 𝗖𝗮𝗿𝗶𝗯𝗯𝗲𝗮𝗻: 𝗶𝗻𝘁𝗲𝗴𝗿𝗮𝘁𝗶𝗼𝗻 𝘃𝗶𝗮 𝗲𝘅𝗽𝗼𝗻𝗲𝗻𝘁𝗶𝗮𝗹 𝘀𝘂𝗯𝘀𝘁𝗶𝘁𝘂𝘁𝗶𝗼𝗻
We introduce 𝗘𝘅𝗽𝗼𝗻𝗲𝗻𝘁𝗶𝗮𝗹 𝗦𝘂𝗯𝘀𝘁𝗶𝘁𝘂𝘁𝗶𝗼𝗻, which is a transformation that simplifies certain composite trigonometric integrals and offers an alternative approach to integrating through trigonometric, hyperbolic, and Euler substitutions. For a more detailed description of how this technique works, visit the blog post
'Integration Using Euler-like Identities' at https://geometriadominicana.blogspot.com/2024/03/integration-using-some-euler-like.html
Examples 2, 3, 7, 8, 9 and 10 illustrate how this method could be 𝗺𝗼𝗿𝗲 𝗲𝗳𝗳𝗶𝗰𝗶𝗲𝗻𝘁 than the traditional approaches.
#math #calculus #integration #newtechnique #complexnumbers #halfangleapproach #symmetrymatters
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I gave this integral as an answer at MathSE.
\[\int_{0}^{1}\left(\frac{5}{2} \left((x - \sqrt{x^2 - 1})^{2i} + x^4\right)-1\right)\,dx=e^\pi.\]
The integral yields \(e^{\pi}\). I don't know why the LaTeX isn't displaying properly.
#math #integral #eulernumber #halfangleapproach #symmetrymatters #calculus
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I gave this integral as an answer at MathSE.
\[\int_{0}^{1}\left(\frac{5}{2} \left((x - \sqrt{x^2 - 1})^{2i} + x^4\right)-1\right)\,dx=e^\pi.\]
The integral yields \(e^{\pi}\). I don't know why the LaTeX isn't displaying properly.
#math #integral #eulernumber #halfangleapproach #symmetrymatters #calculus
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I gave this integral as an answer at MathSE.
\[\int_{0}^{1}\left(\frac{5}{2} \left((x - \sqrt{x^2 - 1})^{2i} + x^4\right)-1\right)\,dx=e^\pi.\]
The integral yields \(e^{\pi}\). I don't know why the LaTeX isn't displaying properly.
#math #integral #eulernumber #halfangleapproach #symmetrymatters #calculus
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I gave this integral as an answer at MathSE.
\[\int_{0}^{1}\left(\frac{5}{2} \left((x - \sqrt{x^2 - 1})^{2i} + x^4\right)-1\right)\,dx=e^\pi.\]
The integral yields \(e^{\pi}\). I don't know why the LaTeX isn't displaying properly.
#math #integral #eulernumber #halfangleapproach #symmetrymatters #calculus
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An 'impossible' integral to solve (Mathematica fails):
\[\int_{2}^{3} \frac{{1 - \tan\frac{{\sec^{-1}x}}{2}}}{{1 + \tan\frac{{\sec^{-1}x}}{2}}}\sqrt{\tan\frac{\csc^{-1}x}{2}} \,dx\]
Unless you know my new technique:
https://mathoverflow.net/questions/463459/identities-that-simplify-tedious-integrals?noredirect=1#comment1203657_463459
#math #impossibleintegral #calculus #halfangleapproach #symmetrymatters -
An 'impossible' integral to solve (Mathematica fails):
\[\int_{2}^{3} \frac{{1 - \tan\frac{{\sec^{-1}x}}{2}}}{{1 + \tan\frac{{\sec^{-1}x}}{2}}}\sqrt{\tan\frac{\csc^{-1}x}{2}} \,dx\]
Unless you know my new technique:
https://mathoverflow.net/questions/463459/identities-that-simplify-tedious-integrals?noredirect=1#comment1203657_463459
#math #impossibleintegral #calculus #halfangleapproach #symmetrymatters -
An 'impossible' integral to solve (Mathematica fails):
\[\int_{2}^{3} \frac{{1 - \tan\frac{{\sec^{-1}x}}{2}}}{{1 + \tan\frac{{\sec^{-1}x}}{2}}}\sqrt{\tan\frac{\csc^{-1}x}{2}} \,dx\]
Unless you know my new technique:
https://mathoverflow.net/questions/463459/identities-that-simplify-tedious-integrals?noredirect=1#comment1203657_463459
#math #impossibleintegral #calculus #halfangleapproach #symmetrymatters -
In my latest blog post, I have derived what appears to be a new formula for solving quadratic equations. Were you aware of it?
Link: https://geometriadominicana.blogspot.com/2024/01/trigonometric-formula-for-solving.html
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In my latest blog post, I have derived what appears to be a new formula for solving quadratic equations. Were you aware of it?
Link: https://geometriadominicana.blogspot.com/2024/01/trigonometric-formula-for-solving.html
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In my latest blog post, I have derived what appears to be a new formula for solving quadratic equations. Were you aware of it?
Link: https://geometriadominicana.blogspot.com/2024/01/trigonometric-formula-for-solving.html
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In my latest blog post, I have derived what appears to be a new formula for solving quadratic equations. Were you aware of it?
Link: https://geometriadominicana.blogspot.com/2024/01/trigonometric-formula-for-solving.html
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In my latest blog post, I have derived what appears to be a new formula for solving quadratic equations. Were you aware of it?
Link: https://geometriadominicana.blogspot.com/2024/01/trigonometric-formula-for-solving.html
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Do you know what I mean, right? Half-angle formulas are central! Their utility for you will depend on how ingenious you are. Look at Euler, learn from him, he is the master of all of us, someone once said.
A proof : https://m.youtube.com/watch?v=eYow30t9aQg&si=S8J4Kk43VgbybtXx&fbclid=IwAR1jf1tM-rgZMapC07vH0W6v7QMQDfM_EXDy_-7p4358XZFGASyZ8gnH1-Q -
Do you know what I mean, right? Half-angle formulas are central! Their utility for you will depend on how ingenious you are. Look at Euler, learn from him, he is the master of all of us, someone once said.
A proof : https://m.youtube.com/watch?v=eYow30t9aQg&si=S8J4Kk43VgbybtXx&fbclid=IwAR1jf1tM-rgZMapC07vH0W6v7QMQDfM_EXDy_-7p4358XZFGASyZ8gnH1-Q -
Do you know what I mean, right? Half-angle formulas are central! Their utility for you will depend on how ingenious you are. Look at Euler, learn from him, he is the master of all of us, someone once said.
A proof : https://m.youtube.com/watch?v=eYow30t9aQg&si=S8J4Kk43VgbybtXx&fbclid=IwAR1jf1tM-rgZMapC07vH0W6v7QMQDfM_EXDy_-7p4358XZFGASyZ8gnH1-Q -
Another theorem falls under the grasp of the half-angle approach: Burlet's theorem.
I give a generalization here: https://geometriadominicana.blogspot.com/2024/01/a-generalization-of-burlets-theorem-to.html?m=1&fbclid=IwAR1oQEwlKoQl2Ajv5BDoXdGHtUUew46WiB9iHqWALL4p1o58I8m8JdMzuT8
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Another theorem falls under the grasp of the half-angle approach: Burlet's theorem.
I give a generalization here: https://geometriadominicana.blogspot.com/2024/01/a-generalization-of-burlets-theorem-to.html?m=1&fbclid=IwAR1oQEwlKoQl2Ajv5BDoXdGHtUUew46WiB9iHqWALL4p1o58I8m8JdMzuT8
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Another theorem falls under the grasp of the half-angle approach: Burlet's theorem.
I give a generalization here: https://geometriadominicana.blogspot.com/2024/01/a-generalization-of-burlets-theorem-to.html?m=1&fbclid=IwAR1oQEwlKoQl2Ajv5BDoXdGHtUUew46WiB9iHqWALL4p1o58I8m8JdMzuT8
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I GOT PUBLISHED!!
The MATINF magazine has officially published my generalization of Newton's formula.
I am immensely grateful to Masters Leo Giugiuc and Costel Balcau for being instrumental in making this publication possible.
You can download and view the article on page 19 at the following link: http://matinf.upit.ro/MATINF9_10/mobile/index.html
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I GOT PUBLISHED!!
The MATINF magazine has officially published my generalization of Newton's formula.
I am immensely grateful to Masters Leo Giugiuc and Costel Balcau for being instrumental in making this publication possible.
You can download and view the article on page 19 at the following link: http://matinf.upit.ro/MATINF9_10/mobile/index.html
-
I GOT PUBLISHED!!
The MATINF magazine has officially published my generalization of Newton's formula.
I am immensely grateful to Masters Leo Giugiuc and Costel Balcau for being instrumental in making this publication possible.
You can download and view the article on page 19 at the following link: http://matinf.upit.ro/MATINF9_10/mobile/index.html
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After discovering the central role that half-angle formulas play in relation to classical metric geometry, the natural question that arises is, why on earth are these formulas so useful? And my answer is symmetry, the same underlying concept in Galois theory or Noether's theorems.
#math #halfangleapproach #symmetrymatters -
After discovering the central role that half-angle formulas play in relation to classical metric geometry, the natural question that arises is, why on earth are these formulas so useful? And my answer is symmetry, the same underlying concept in Galois theory or Noether's theorems.
#math #halfangleapproach #symmetrymatters -
After discovering the central role that half-angle formulas play in relation to classical metric geometry, the natural question that arises is, why on earth are these formulas so useful? And my answer is symmetry, the same underlying concept in Galois theory or Noether's theorems.
#math #halfangleapproach #symmetrymatters -
After discovering the central role that half-angle formulas play in relation to classical metric geometry, the natural question that arises is, why on earth are these formulas so useful? And my answer is symmetry, the same underlying concept in Galois theory or Noether's theorems.
#math #halfangleapproach #symmetrymatters -
After discovering the central role that half-angle formulas play in relation to classical metric geometry, the natural question that arises is, why on earth are these formulas so useful? And my answer is symmetry, the same underlying concept in Galois theory or Noether's theorems.
#math #halfangleapproach #symmetrymatters