Separating real and complex parts of a number

[KarolisK]

Homework Statement

Hello, I am supposed to express the and the phase part of expression:

$\displaystyle{S=\frac{k}{\sqrt{1+i\gamma_0}} \cdot exp\left(\frac{z}{1 + i\gamma_0}\right)}$

Homework Equations

The answer should be in the form:

$\displaystyle{S=a(\gamma_0) \cdot exp\left(i\varphi(\gamma_0)\right)}$

The Attempt at a Solution

Well, its clear(probably) that for the amplitude part I just have to multiply this equation by its complex conjugate and take a square root out of the result. This leaves me with the expression of:
$\displaystyle{ a(\gamma_0)=\frac{k}{\left(1+\gamma^2_0\right)^{1/4}} \cdot exp\left(\frac{z}{1+\gamma^2_0}\right) }$

However, I don't quite understand how to get the complex(phase) part of the number? A hint where to start would be very gladly accepted :), thank you.

 

[ehild]

]
Well, its clear(probably) that for the real part I just have to multiply this equation by its complex conjugate and take a square root out of the result.

It is the magnitude or absolute value of the complex number instead of the real part. So how do you get the real and imaginary parts?


ehild

 

[KarolisK]

Ah yes, sorry, its the amplitude, thanks for noticing. I'll reformulate the problem. I need to get the expressions for the amplitude and the phase.

 

[ehild]

Is z in the exponent a complex number? Then write it out with its real and imaginary parts, and find the real and imaginary parts of z/(1+iγ₀), so you have the exponential in the form exp(u+iv) .

Write the fraction 1/(1+iγ₀) in exponential form, too: exp(a+ib). Then your formula is equal to k*exp((a+u)+i(b+v)). k*exp(a+b) is the magnitude, and the phase is b+v.

ehild

 

[KarolisK]

z is real and negative and k is real and positive constant. Anyway, expressing the fraction sqrt(1/(1+iy0) can get me just as close as:

$\displaystyle{ k \cdot exp \left(\frac{1}{2}ln \left( \frac{1}{1+\gamma^2_0}-\frac{i\gamma_0}{1+\gamma^2_0}\right) \right) }$

Which I don't understand how to simplify to form exp(a+ib). I have also tried expressing phase from the general form:

$\displaystyle{ \frac{S}{a(\gamma_0)}=exp \left(i \varphi \right) }$

Which is kinnda closer to the answer with the expression:

$\displaystyle{ i\varphi=\frac{1}{4}ln\left(\frac{1-i\gamma_0}{1+i\gamma_0} \right) - \frac{i\gamma_0z}{1+\gamma^2_0} }$

The answer should be:

$\displaystyle{\varphi = -\frac{1}{2}arctan\ \gamma_0 - \frac{\gamma_0z}{1+\gamma^2_0} }$

 

[ehild]

Write all factors in the exponential form: r*e^iφ. Any complex number u+iv= r*e^iφ, where $r=\sqrt{(u^2+v^2) }$ and tan(φ)=v/u.
This way, $1+i \gamma_0=\sqrt{1+\gamma_0 ^2} e^ {i\arctan(\gamma_0)}$

ehild

 

[KarolisK]

ah yes, thank you very much:)