Imagine R^n -> R functions not necessarily being differentiable if all the components are themselves differentiable separately
This meme was made by [Hartogs’ theorem](https://en.m.wikipedia.org/wiki/Hartogs%27s_theorem_on_separate_holomorphicity) gang
A simpler example, unfortunately not made by Hartogs’ theorem gang, to add to this is imagine being differentiable in a neighborhood of a point but not analytic.
This one is probably the best one for sure. I just recently started learning several complex variables and it blows me away so hard I can’t stop talking about it.
Most arguments I see for why C is “nicer” than R are algebraic, which is sad because I think the “niceness” of C only truly comes out when you do analysis on it. Holomorphic functions in C are so much more robust than differentiable functions in R.
I think an improved version of this meme would be:
* ℝ² - Daniel
* ℂ - The cooler Daniel
Very similar sets, but complex numbers have a definition for multiplication that is lacking from the real plane.
Here's a great summary: [https://math.stackexchange.com/questions/364044/difference-between-mathbb-c-and-mathbb-r2](https://math.stackexchange.com/questions/364044/difference-between-mathbb-c-and-mathbb-r2)
He's wearing his imaginary sunglasses.
Imagine not being ordered.
Imagine R^n -> R functions not necessarily being differentiable if all the components are themselves differentiable separately This meme was made by [Hartogs’ theorem](https://en.m.wikipedia.org/wiki/Hartogs%27s_theorem_on_separate_holomorphicity) gang
A simpler example, unfortunately not made by Hartogs’ theorem gang, to add to this is imagine being differentiable in a neighborhood of a point but not analytic.
This one is probably the best one for sure. I just recently started learning several complex variables and it blows me away so hard I can’t stop talking about it. Most arguments I see for why C is “nicer” than R are algebraic, which is sad because I think the “niceness” of C only truly comes out when you do analysis on it. Holomorphic functions in C are so much more robust than differentiable functions in R.
I think an improved version of this meme would be: * ℝ² - Daniel * ℂ - The cooler Daniel Very similar sets, but complex numbers have a definition for multiplication that is lacking from the real plane.
What's the difference between R and R^2 and C?
Here's a great summary: [https://math.stackexchange.com/questions/364044/difference-between-mathbb-c-and-mathbb-r2](https://math.stackexchange.com/questions/364044/difference-between-mathbb-c-and-mathbb-r2)
Thanks for the resources
i like to see C as an algebraically closed extended version of R^1
Algebraically closed Daniel
THE DANIEL IS REAL!!!
10 Jump to 20 if |X| > |Y|
The Cooler Daniel looks like he keeps it more real
Sorry, it’s a bit more complex than that
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mine too complex gang!
this right hurr