- #1
kasse
- 384
- 1
z^4= 1/2 + i sqrt(3)/2
I start by transforming into polar form:
z^4 = e^(i*Pi/3)
But then I'm blank.
I start by transforming into polar form:
z^4 = e^(i*Pi/3)
But then I'm blank.
kasse said:z^4= 1/2 + i sqrt(3)/2
I start by transforming into polar form:
z^4 = e^(i*Pi/3)
But then I'm blank.
CompuChip said:I believe the complex root is defined by (that is: it's usually continued from the real function by)
[tex] \sqrt{r e^{i \phi}} = \sqrt{r} e^{i \phi / 2} [/tex]
malawi_glenn said:yes, and from De Moivre's formula we get the general:
[tex] z^{1/n} = r^{1/n}*exp(i \phi / n) [/tex]
The root of a complex number is a number that, when multiplied by itself a certain number of times, gives the original complex number. It is the inverse operation of exponentiation.
To find the root of a complex number, you can use the formula r = a1/n * (cos(θ + 2kπ) + i * sin(θ + 2kπ)), where r is the magnitude of the complex number, a is the base of the root, n is the index of the root, θ is the argument of the complex number, and k is an integer representing the different roots.
A complex number has n distinct nth roots, where n is the index of the root. This means that for every complex number, there are n different answers when finding its nth root.
The principal root of a complex number is the root with the smallest positive argument. It is commonly denoted as r1/n, where r is the magnitude of the complex number and n is the index of the root.
The root of a complex number is closely related to De Moivre's theorem, which states that (cos(θ) + i * sin(θ))n = cos(nθ) + i * sin(nθ). This theorem can be used to find the roots of a complex number by setting n equal to the index of the root.