Complex Numbers - help

[alwingeorge]

If z^2 = a+bi, where a∈R and b∈R find all the solutions for z. Hint: let z = x+yi (Here i stands for √−1 and R stands for real numbers)

I have no idea on how to do this. Any help would be appreciated. Thank you in advance

 

[stapel]

If z^2 = a+bi, where a∈R and b∈R find all the solutions for z. Hint: let z = x+yi (Here i stands for √−1 and R stands for real numbers)

I have no idea on how to do this. Any help would be appreciated. Thank you in advance

I'm not clear myself on quite what is being asked. Since you have no values for a and b, how is one to obtain numerical values for z? Or is this maybe asking for z in terms of a and b, given that the square of z is equal to some complex number a + bi?

If so, then:

z^2 = (x + yi)^2 = x^2 + 2xyi - y^2 = (x^2 - y^2) + (2xy)i = a + bi

Maybe do something with this...?

 

[alwingeorge]

I'm not clear myself on quite what is being asked. Since you have no values for a and b, how is one to obtain numerical values for z? Or is this maybe asking for z in terms of a and b, given that the square of z is equal to some complex number a + bi?

If so, then:

z^2 = (x + yi)^2 = x^2 + 2xyi - y^2 = (x^2 - y^2) + (2xy)i = a + bi

Maybe do something with this...?

The question is asking for z in terms of a and b and tank you for your input btw.

 

[Deleted member 4993]

The question is asking for z in terms of a and b and tank you for your input btw.

Using the hint above and using De Moivre's formula, express \(\displaystyle z = \sqrt{z^2} = \sqrt{a+ib} = ? \).

 

[HallsofIvy]

If z^2 = a+bi, where a∈R and b∈R find all the solutions for z. Hint: let z = x+yi (Here i stands for √−1 and R stands for real numbers)

I have no idea on how to do this. Any help would be appreciated. Thank you in advance

As the problem says, let z= x+ iy. Then \(\displaystyle z^2= (x+ iy)^2= (x^2- y^2)+ 2xyi= a+ bi\). Solve the equations x^2- y^2= a and 2xy= b.

 

[alwingeorge]

As the problem says, let z= x+ iy. Then \(\displaystyle z^2= (x+ iy)^2= (x^2- y^2)+ 2xyi= a+ bi\). Solve the equations x^2- y^2= a and 2xy= b.

Thank you for your help, I will work with that.