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Where is the mistake?

I know that log function is defined only for positive reals but please help me understand where i go wrong in the following proof : TO PROVE : $\ln\big(i) = \frac{i\pi}{2}$ PROOF : $\ln(i) = \ln(e^{\frac{i\pi}{2}})$ but we know that $\ln(e^x) = x$ and hence $\ln(i) = \ln(e^{\frac{i\pi}{2}}) = \frac{i\pi}{2}$ Hence proved . Is it correct ? Although , i am sure it is wrong but which step is precisely not allowed here?

Note by Starwar Clone
1 year ago

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it's not the same the exponentation or logharythmic functions with the complex numbers, that the exponentation or logharytmic functions with real numbers. Look at this : $$i = e^{ \frac {i \pi}{2}} =e^{ \frac {5i \pi}{2}}$$...

- 1 year ago

thanks for the help :)

- 1 year ago

- 1 year ago

thank you ...

- 1 year ago

thank you for you,of course

- 1 year ago

this discussion has turned out to be fun......all are becoming friends :)

- 1 year ago

thank you again friend... I like that we are all friends

- 1 year ago

Hello my friend again

- 1 year ago

What makes you think that $$\ln i \neq \frac{ i \pi } { 2}$$?

Also, be very careful with complex exponentiation and logarithms.

Staff - 1 year ago

The sequence of steps that you took essentially work, but you have to be clear if you are working with "equality of multi-valued functions" or "evaluate at primitive branch".

To be clear, the multi-valued function $$\ln i$$ is indeed equal to $$\frac{ i \pi }{2} + 2k \pi i$$.

Staff - 1 year ago

Calvin, please, I want to be your friend, but you don't make me easy it... $$\frac{i \pi}{2} + 2k\pi \neq i\cdot(\frac{\pi}{2} + 2k \pi)$$

- 1 year ago

Ah yes, fixed the typo. thanks.

Staff - 1 year ago

Don't worry, can we be friends?

- 1 year ago

Of course!

Staff - 1 year ago

Thank you very much.

- 1 year ago

Calvin, I would write great wikis, but I have some problems:

a) Where can I start?

b) I can't do everything, I'm not God/Godness or...

c) For example, suppose I want to write a wiki on numerical analysis, before doing this, would I have to define what a vectorial space is, would I have to give a topology or abstract algebra course first, would I have to establish some prior knowledge of mathematical analysis?...

d) In short, what would I have to do?

- 1 year ago

1. Here is a list of wikis that we're working on. If there are any that you're interested in helping out with, ping #wikis.
If you're working on a new wiki and need help with developing the outline, let me know.

2. Don't worry about it. Focus on what you're good at / what you're interested in.

3. Do not worry about ensuring that foundational material exists when developing out a page. We can fill that in subsequently. and use "red links" to indicate that wiki pages are undeveloped. E.g. In the Gram Schmidt wiki, we would assume knowledge of vector space, and link out to it.

4. Where do you want to start?

Staff - 1 year ago

Good answer, let's start at the beginning, there is already a story about this, a teacher from Tai-chi taught it to me. Suppose I want to talk about arithmetic theory: where do I have to start? Defining Peano's axioms?... Now suppose I want to talk about set theory, would I have to give a theory about what a class is?

P.S.- Imagine that I am nobody, neither a mathematician nor a physicist ..

- 1 year ago

Look, we can make one thing, I like so much topology, and here I don't see Topology. What about if I start talking about Topology, with more or less a notion of a "set", I'll "definite it", I don't want Russell's paradox again... do you remember what happened...? and later I continue talking about Topology, and when I need help, I can email you or stop... Remember, we are free if we respect ourselves... what do you think about this?... I have other question: where is Mateo Matija...? I can't get see him... It's important for me...

- 1 year ago

- 1 year ago

Yes, I'm going on there...

- 1 year ago

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