Muphrid said:For integer [itex]k[/itex], sure they are. Any full turn around the unit circle doesn't change the value.
Muphrid said:Oh, okay, I see what's going on. Yeah, in this case, you may be more correct than the book is. Someone with more recent experience than I in complex analysis may know something more specific to this, though.
Aziza said:Write the following in x+iy form: (-1)^i
my solution:
(-1)i = (ei(π+2πk))i = e-(π+2πk)
However, my book just states the answer as e-(π) ...but these are not the same...
(Those π's are hard to read !)Aziza said:Write the following in x+iy form: (-1)^i
my solution:
(-1)i = (ei(π+2πk))i = e-(π+2πk)
However, my book just states the answer as e-(π) ...but these are not the same...
A "Very simple complex powers problem" is a mathematical problem that involves raising a complex number to a certain power. Complex numbers are numbers that have both a real and imaginary component, and they are represented in the form a + bi, where a and b are real numbers and i is the imaginary unit. Raising a complex number to a power involves multiplying it by itself a certain number of times.
To solve a "Very simple complex powers problem", you can use De Moivre's theorem, which states that for any complex number z = r(cosθ + isinθ), the nth power of z is equal to r^n(cos(nθ) + isin(nθ)). This allows you to easily calculate the result of raising a complex number to a power without having to carry out multiple multiplications.
"Very simple complex powers problems" have many real-world applications, particularly in fields such as electrical engineering and physics. They can be used to model and analyze the behavior of electrical circuits, as well as in the study of oscillations and waves.
One common challenge in solving "Very simple complex powers problems" is understanding the concept of complex numbers and how they behave when raised to a power. Another challenge may be keeping track of the different steps involved in solving the problem, such as using De Moivre's theorem or converting between polar and rectangular forms of complex numbers.
Yes, to solve "Very simple complex powers problems" more efficiently, it is important to have a strong understanding of complex numbers and the properties of their powers. It can also be helpful to practice converting between polar and rectangular forms and familiarizing yourself with De Moivre's theorem. Additionally, breaking the problem down into smaller steps and keeping organized notes can also make the process more efficient.