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shapis@lemmy.ml ⁨6⁩ ⁨days⁩ ago

This very nice Romanian lady that taught me complex plane calculus made sure to emphasize that e^j*theta was just a notation.

Then proceeded to just use it as if it was actually eulers number to the j arg. And I still don’t understand why and under what cases I can’t just assume it’s the actual thing.

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frezik@lemmy.blahaj.zone ⁨6⁩ ⁨days⁩ ago
Let’s face it: Calculus notation is a mess. We have three different ways to notate a derivative, and they all suck.

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- JackbyDev@programming.dev ⁨5⁩ ⁨days⁩ ago
  Calculus was the only class I failed in college. It was one of those massive 200 student classes. The teacher had a thick accent and hand writing that was difficult to read. Also, I remember her using phrases like “iff” that at the time I thought was her misspelling something only to later realize it was short hand for “if and only if”, so I can’t imagine how many other things just blew over my head.
  
  I retook it in a much smaller class and had a much better time.
  
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zea_64@lemmy.blahaj.zone ⁨6⁩ ⁨days⁩ ago
I’ve seen e^{d/dx}

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jsomae@lemmy.ml ⁨5⁩ ⁨days⁩ ago
e^iθ^ is not just notation. You can graph the entire function e^x + iθ^ across the whole complex domain and find that it matches up smoothly with both the version restricted to the real axis (e^x^) and the imaginary axis (e^iθ^)

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- shapis@lemmy.ml ⁨5⁩ ⁨days⁩ ago
  I’m aware of that identity. There’s a good chance I misunderstood what she said about it being just a notation.
  
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  - jsomae@lemmy.ml ⁨4⁩ ⁨days⁩ ago
    It’s not simply notation, since you can prove the identity from base principles. An alien species would be able to discover this independently.
    
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sabin@lemmy.world ⁨5⁩ ⁨days⁩ ago
It legitimately IS exponentiation. Romanian lady was wrong.

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carmo55@lemmy.zip ⁨5⁩ ⁨days⁩ ago
It is just a definition, but it’s the only definition of the complex exponential function which is well behaved and is equal to the real variable function on the real line.

Also, every identity about analytical functions on the real line also holds for the respective complex function (excluding things that require ordering). They should have probably explained it.

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- shapis@lemmy.ml ⁨5⁩ ⁨days⁩ ago
  She did. She spent a whole class on about the fundamental theorem of algebra I believe? I was distracted though.
  
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