Domains of some functions

[pulkit]

I was just curious to know wether the trignometric functions and the log / antilog functions are defined for imaginary and complex numbers....and if yes then how are they defined ?

[ed84c]

Euler wrote a piece on this titled "De la Controverse entre Mrs Leibniz

et Bernoulli sur les Logarithmes des Nombres Negatifs et Imaginaires",

 

Euler proves that each nonzero real number x has an infinity of logarithms. If x is positive then all but one of the logarithms are imaginary, if x is negative then all the logarithms are imaginary. He does this by use of the equation log(x) = nx^(1/n)-n, which is true when n is infinitely large, and familiar properties of the number of roots x^(1/n) and their properties for finite n. But i dont know about trig with imaginaries. I do no you can map time with Is tho with some 'interesing effects' acording to mr Hawking.

[bloodhound]

well it can be shown inverse sin or inverse cos of numbers outside [-1,1] gives rise to complex numbers. so yes some trig functions are defined for complex numbers

 

you such use the addition and subraction formula

 

sin(a+ib)= blah blah blah. and then use the fact that some part of them is equal to the hyperbolic functions due to the i being involved. i cant remember most of the stuff.

[e(ho0n3]

I was just curious to know wether the trignometric functions and the log / antilog functions are defined for imaginary and complex numbers....and if yes then how are they defined ?

Yes and yes. Examples: sin(a + ib) = sin(a)cosh(b) + icos(a)sinh(b). It's a bit trickier with logarithms.

[pulkit]

What if you took the series expansion of the function (MacLauren series), and plugged the complex/imaginary number ............

 

I think thats a way to end up with e^(i theta) = cos (theta) + i* sin (theta)

[ed84c]

interesting idea

[Dave]

What if you took the series expansion of the function (MacLauren series)

 

Which function? sine and cosine?

[pulkit]

Yes sine,cosine and tan .

I supose it wouldn't work ion log because there are some conditions on the values that you can put into the expansion of a logarithmic function

[Dave]

What if you took the series expansion of the function (MacLauren series)' date=' and plugged the complex/imaginary number ............

 

I think thats a way to end up with e^(i theta) = cos (theta) + i* sin (theta)[/quote']

 

The way you come up with that is by expanding [math]e^{i\theta}[/math], and then noticing that you can seperate the real terms and the imaginary terms into two seperate power series; one for cosine and one for sine.

[pulkit]

The way you come up with that is by expanding e^(i*theta), and then noticing that you can seperate the real terms and the imaginary terms into two seperate power series; one for cosine and one for sine.

 

What if you similarly expand say Sin(a+ib) and then try to solve the resulting terms or just try to sum them up?