Is there a relation between log and arcs for complex numbers?

[Jhenrique]

If there is a formula relating the exponential with sine and cosine normal and hyperbolic (exp(ix) = cos(x) + i sin(x), exp(x) = cosh(x) + sinh(x)), there is also a formula relating the logarithm with arcsin, arccos, and arcsinh arccosh?

 

[SteamKing]

Logs are implicitly in the definition of arcsinh and arccosh:

http://en.wikipedia.org/wiki/Hyperbolic_function

For arcsin and arccos, the logarithms would have imaginary arguments.

 

[Jhenrique]

But, But I wonder if there is a general expression that combines the sine the cosine (hyperbolic or no) of one side of the equality with the logarithm, in other side of the equality...

 

[SteamKing]

Are you talking about expressing sin(x) and cos(x) in terms of log(x)? Your question is not very clear.

 

[Jhenrique]

Sorry. I'm talking about an expression of log(x) in terms of arcsinh(x) and arcosh(x).

 

[SteamKing]

Already answered in Post #2

 

[Jhenrique]

Hmm ... let me I explain me better...

http://imageshack.com/a/img513/8899/5cgc.png

You agree with me that is missing two equations?

 

[SteamKing]

You mean ln (e^x) = x and ln (e^ix) = ix? I'm sorry, I'm not following your question.

 

[Jhenrique]

I apologize too, because my English is primitive...

I'm trying to say that if you can combine sine and cosine to express the exponential, then it should also be possible to combine and arcsin arccos to express the logarithm. But this combination is not so simple ... I tried to add and multiply arcsineh(x) with arccosh(x), I tried to combine they by arithmetic and geometric mean, I tried to break log(x) on even and odd function. I've tried several things, but I was not able to find a true expression.

I look for an expression as log(x) = arccosh(x) + arcsinh(x). This expression is false, but it is close of the genuine.

 

[lurflurf]

Do you mean

$$\log(x)=\mathrm{arcsinh} \left( \frac{x^2-1}{2x} \right)=\imath \arcsin \left( \frac{x^2-1}{2 \imath x} \right)$$

This only holds for 0<x
but similar expressions can be used for x complex