Complex coordinate space of dimension 2 by 
Ciro Santilli 35 Updated 2025-04-16 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-04-16 +Created 1970-01-01This section is about the definition of the dot product over , which extends the definition of the dot product over .
Some motivation is discussed at: math.stackexchange.com/questions/2459814/what-is-the-dot-product-of-complex-vectors/4300169#4300169
The complex dot product is defined as:
E.g. in :
Just like the usual dot product, this will be a positive definite symmetric bilinear form by definition.
Norm induced by the complex dot product by Ciro Santilli 35 Updated 2025-04-16 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-04-16 +Created 1970-01-01Each elliptic space can be modelled with a real projective space. The best thing is to just start thinking about the real projective plane.
Complex Analysis by Juan Carlos Ponce Campuzano by Ciro Santilli 35 Updated 2025-04-16 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-04-16 +Created 1970-01-01Generalization of Laplace's equation where the value is not necessarily 0.
Being a complex holomorphic function is an extremely strong condition.
The existence of the first derivative implies the existence of all derivatives.
Another extremely strong consequence is the identity theorem.
"Holos" means "entire" in Greek, so maybe this is a reference to the fact that due to the identity theorem, knowing the function on a small open ball implies knowing the function everywhere.
visualizing the Riemann hypothesis and analytic continuation by 3Blue1Brown (2016) is a good quick visual non-mathematical introduction is to it.
The key question is: how can this continuation be unique since we are defining the function outside of its original domain?
The answer is: due to the identity theorem.
Visualizing the Riemann hypothesis and analytic continuation by 3Blue1Brown (2016) by Ciro Santilli 35 Updated 2025-04-16 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-04-16 +Created 1970-01-01The 3D structure of GFP is so cool. It is so clearly a bottle with a fluorescent bit well isolated right in the middle. Like a little lamp.
Sample software implementations:
Order of the highest derivative that appears.
Technique to solve partial differential equations
Naturally leads to the Fourier series, see: solving partial differential equations with the Fourier series, and to other analogous expansions:
One notable application is the solution of the Schrödinger equation via the time-independent Schrödinger equation.
Numerical method to solve a partial differential equation by Ciro Santilli 35 Updated 2025-04-16 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-04-16 +Created 1970-01-01The finite element method is one of the most common ways to solve PDEs in practice.
Pinned article: ourbigbook/introduction-to-the-ourbigbook-project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
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This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivative - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
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Figure 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.
Figure 3. Visual Studio Code extension installation.
Figure 4. Visual Studio Code extension tree navigation.
Figure 5. . You can also edit articles on the Web editor without installing anything locally. Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.Video 4. OurBigBook Visual Studio Code extension editing and navigation demo. Source. - Internal cross file references done right:
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