What Is the Form of G When k Is Complex?

In summary, a complex complex root is a solution of a polynomial equation with complex coefficients that results in a value of zero when plugged in. It differs from a regular complex root in that it requires two imaginary components. To determine the complex complex roots, one can use the quadratic formula or other methods such as factoring or graphing. A polynomial equation can have both real and complex complex roots, and understanding these roots has practical applications in various fields such as signal processing, electrical engineering, physics, cryptography, and computer science.
  • #1
madeinmsia
15
0
G'' = -kG
k is a constant
solving this ODE, r = +/- sqrt(-k)
if k > 0, then r = +/- sqrt(k)i
so G is in the form Acos(sqrt(k)x) + Bsin(sqrt(k)x)

so, what if k is a complex number, then
r = +/- sqrt(-ki)

then what is the form of G?
 
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  • #2
madeinmsia said:
so, what if k is a complex number, then
r = +/- sqrt(-ki)

then what is the form of G?

Don't you know how to calculate http://www.mathpropress.com/stan/bibliography/complexSquareRoot.pdf?
 

FAQ: What Is the Form of G When k Is Complex?

What is a complex complex root?

A complex complex root is a type of mathematical concept that refers to a solution of a polynomial equation with complex coefficients. It is a complex number that, when plugged into the polynomial equation, results in a value of zero.

How is a complex complex root different from a regular complex root?

A complex complex root is different from a regular complex root in that it requires two imaginary components to form a complex number, whereas a regular complex root only requires one imaginary component.

How do you determine the complex complex roots of a polynomial equation?

To determine the complex complex roots of a polynomial equation, you can use the quadratic formula or other methods such as factoring or graphing. It is important to note that the number of complex complex roots will always match the degree of the polynomial equation.

Can a polynomial equation have both real and complex complex roots?

Yes, a polynomial equation can have both real and complex complex roots. In fact, it is common for polynomial equations to have a mix of both real and complex roots.

What are the practical applications of understanding complex complex roots?

Understanding complex complex roots is important in various fields of science and engineering, such as signal processing, electrical engineering, and physics. It also has applications in cryptography and computer science.

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