What is the real part of a function like f(z)= √(z)?

[cj]

I'm used to seeing a function like:

$$\textbf{f}=x+iy \text$$

where

$$i = \sqrt{-1}$$

and understanding the the real part is:

$$\text{Re[f]} = x = Acos\theta$$

What, though, is the real part of a function like, for example,

$$\textbf{f}=\sqrt{x+iy}$$

??

 

[selfAdjoint]

Expand in Taylor's series, paying attention to powers of i, and separate terms with i out to get two series for the real and imaginary parts. This only works where the TS converges absolutely. Where is that?

 

[robphy]

$$Re(f)=\frac{f+f^*}{2}$$
where $$f^*$$ is the complex-conjugate of f.

So, $$\begin{align*} Re(\sqrt{x+iy})&=\frac{(\sqrt{x+iy})+(\sqrt{x+iy})^*}{2}\\ &=\frac{\sqrt{x+iy}+\sqrt{x-iy}}{2} \end{align*}$$

Additionally, you could write $$x+iy$$ in polar coordinates.
Then, do the above.

 

[uart]

Since it looks like you're assuming the x and y are real and using the notation x = A cos(theta), y= A sin(theta) then the answer is just,

sqrt(A) cos(theta / 2)

 

[HallsofIvy]

You need to be more careful in distinguishing between the "real and imaginary parts" of the variable z= x+ iy, and the "real and imaginary parts" of the function value. Most textbooks write f(z)= u+ iv so that u is the real part and v the imaginary part of f: is z= x+ iy then u(x,y) and v(x,y) are real valued functions of the real variables of x and y.

 

[cj]

Since it looks like you're assuming the x and y are real and using the notation x = A cos(theta), y= A sin(theta) then the answer is just,

sqrt(A) cos(theta / 2)

If I am (and I am) going to use polar forms, the
general function:

$$\textbf{f}=x+iy$$

has the magnitude:

$$|\textbf{f}|=\sqrt{x^2+y^2}$$

with

$$x = Acos\theta \text{ and } y = Asin\theta$$

$$\text{BUT, I have the form: } \textbf{f}=\sqrt{x+iy} \text{ , NOT }\textbf{f}=x+iy$$

So how do I make use of

$$|\textbf{f}|=\sqrt{x^2+y^2}$$

since I don't have a function in the form of
$$\textbf{f}=x+iy$$

??

 

[HallsofIvy]

Have you simply ignored what everyone has said?

"I don't have a function in the form of f= x+ iy" is just saying you don't have the identity function: f(z)= z.

You're not going to be able to do very much is f(z)= z is the only function you know how to work with!

Suppose f(z)= z². Then, writing z= x+iy, f(z)= (x+ iy)²=
x²+ 2ixy+ y²(i²)= (x²- y²)+ (2xy)i.
The real part of f is x²- y² and the imaginary part of f is 2xy.

f(z)= √(z) is a little harder just because it is harder to calculate the value. It would probably be best to write z in polar form:
z= x+iy= r(cos(θ)+ i sin(θ)). The f(z)= √(z)= r^1/2(cos(θ/2)+ i sin(θ/2)). The real part of f is r^1/2cos(θ/2) and the real part is r^1/2sin(θ/2).

 

[cj]

Got it -- thanks!

Although I think you meant "imaginary" rather than "real"
in the following:

... and the real part is r^1/2sin(θ/2).

Have you simply ignored what everyone has said?

"I don't have a function in the form of f= x+ iy" is just saying you don't have the identity function: f(z)= z.

You're not going to be able to do very much is f(z)= z is the only function you know how to work with!

Suppose f(z)= z². Then, writing z= x+iy, f(z)= (x+ iy)²=
x²+ 2ixy+ y²(i²)= (x²- y²)+ (2xy)i.
The real part of f is x²- y² and the imaginary part of f is 2xy.

f(z)= √(z) is a little harder just because it is harder to calculate the value. It would probably be best to write z in polar form:
z= x+iy= r(cos(θ)+ i sin(θ)). The f(z)= √(z)= r^1/2(cos(θ/2)+ i sin(θ/2)). The real part of f is r^1/2cos(θ/2) and the real part is r^1/2sin(θ/2).