According to Euler's equation,

 

(e)^ix = cos x + i sin x

 

But, what is the value of any number raised to i ? (i=sqrt(-1))

Obviously, if we already know that "[math]e^{ix}= cos(x)+ i sin(x)[/math]" then, taking x= 1, [math]e^i= cos(1)+ i sin(1)[/math]. So what you are really asking is "how do we get that formula?"

 

Typically, the way we extend function defined on real numbers to the complex number system (or to matrices or other systems in which we have the basic arithmetic operations defined) is to use the Taylor's series for the function.

 

In particular, [math]e^x= \sum \frac{1}{n!} x^n= 1+ x+ \frac{1}{2}x^2+ \frac{1}{3!} x^3+ \cdot\cdot\cdot+ \frac{1}{n!}x^n+ \cdot\cdot\cdot[/math]. Taking x= i, [math]i^2= -1[/math], [math]i^3= (i^2)i= -i[/math], [math]i^4= (i^3)(i)= (-i)(i)= -(-1)= 1[/math] and it all starts again so that becomes [math]e^{ix}= 1+ ix- \frac{1}{2}x^2- \frac{1}{6}ix^3+ \frac{1}{4!}x^4+ \cdot\cdot\cdot[/math]. Of course, all of the even powers of i are either 1 or -1 and all of the odd powers of i are either i or -i. Separating the "real" and "imaginary" parts, [math]e^i= (1- \frac{1}{2}x^2+ \frac{1}{4!}x^4+ \cdot\cdot\cdot + \frac{(-1)^i}{(2i)!x^{2i}\cdot\cdot\cdot)+ i(x- \frac{1}{3!}x^3+ \frac{1}{5!}x^5+ \cdot\cdot\cdot+ frac{(-1)^j}{(2j+1)!}x^{2j+1}+ \cdot\cdot\cdot)[/math] where I have written the even powers as "2i" and the odd powers as "2j+ 1".

 

Now, we recognize the "[math]1- \frac{1}{2}x^2+ \frac{1}{4!}x^4+ \cdot\cdot\cdot + \frac{(-1)^i}{(2i)!}x^{2i}\cdot\cdot\cdot)[/math]" is the Taylor's series for "cos(x)" and "[math]x- \frac{1}{3!}x^3+ \frac{1}{5!}x^5+ \cdot\cdot\cdot+ frac{(-1)^j}{(2j+1)!}x^{2j+1}+ \cdot\cdot\cdot)[/math]" so that [math]e^{ix}= cos(x)+ i sin(x)[/math] and, as I pointed out before, [math]e^i= cos(1)+ i sin(1)[/math] which is, approximately, [math]e^i= 0.5403+ 0.8415i[/math].

OK.Thanks.

According to Euler's equation,

 

(e)^ix = cos x + i sin x

 

But, what is the value of any number raised to i ? (i=sqrt(-1))

 

 

this is not only one theorem or formula of Euler's

 

actually I could not understand clearly what you are asking ..

 

but it would be helpful to check these contexts

 

1) laurent series

2) liouville theorem's

3) C-R equations.

 

I mean you will be able to fgind some relevant informations ,especially "Laurent series " is being used in case of that formula.