It might be more clear in the trigonometric form where the equation iscomplexhuman said:ok...this was meant to be a fun problem :grumpy: but looks like I don't deserve to have fun!
How am I meant to derive trig identities like sin(x)cos^3(x) from some complex **** like [tex]\left( {e}^{{\it ix}} \right) ^{n}={e}^{{\it ixn}}[/tex]! I just don't get the idea!
The definition of (e^ix)^n is the result of raising the complex exponential number e^ix to the power of n. This means that the number e^ix is multiplied by itself n times.
The difference between (e^ix)^n and e^(ixn) lies in the order of operations. In (e^ix)^n, the complex exponential number e^ix is raised to the power of n first, and then multiplied by itself n times. In e^(ixn), the complex exponential number is multiplied by n first, and then raised to the power of ix. This can result in different numerical values.
(e^ix)^n can be simplified using the trigonometric identity e^(ix) = cos(x) + i*sin(x). This means that (e^ix)^n can be written as (cos(x) + i*sin(x))^n. From there, the binomial theorem can be used to expand and simplify the expression.
(e^ix)^n is commonly used in scientific calculations involving oscillations and waves, such as in electrical engineering, physics, and signal processing. It is also used in the study of complex numbers and their applications in various fields of science and mathematics.
Yes, (e^ix)^n can be used to solve real-world problems that involve oscillations and waves, such as in the analysis of electrical circuits and the behavior of electromagnetic waves. It is also used in the study of quantum mechanics and other branches of physics.