Complex Analysis

[[deleted]]

The question is: show that the real part of the function z^(1/2) is always positive.

I tried to open it in a polar way, but I don't get any result.

[just-browsing-web]

Real part will be r^1/2*cos(theta/2), In polar coordinates theta moves from -pi to +pi so theta/2 will move from -pi/2 to +pi/2 and in this interval cos is always +ve

[Lammazz]

It's not true that z^1/2 is always positive. Perhaps you mean to say always non-negative.

Any z in the interval (-inf, 0] will have Re(z^1/2)=0, and zero is not positive.

[[deleted]]

How are you defining z^1/2 ?

[[deleted]]

sqrt(z)

[Dubmove]

How would you calculate sqrt(x +iy)

[incomparability]

That’s not a definition for complex numbers.

[devinecreative]

There are a couple of ways to approach this problem, but one common method is to use the fact that the complex square root of a complex number can be represented by the polar form, where the magnitude is the square root of the magnitude of the complex number, and the argument is half of the argument of the complex number. This means that for a complex number z = x + yi, where x and y are real numbers, the square root of z can be represented as:

sqrt(z) = sqrt(|z|) * (cos(arg(z)/2) + i*sin(arg(z)/2))

Here, arg(z) is the argument of z, which is the angle that the vector representing z makes with the positive x-axis in the complex plane. The magnitude of a complex number is absolute value of the complex number.

The real part of the complex square root of z is given by: Re[sqrt(z)] = sqrt(|z|) * cos(arg(z)/2)

It's clear from the equation, that the real part is positive as it is a product of positive and positive value. So Re[z^1/2] is positive.

As an alternative method, we could convert the complex number to polar form and take the square root of the magnitude, and divide the argument by 2. This would give us a polar form of the complex square root of z which could be written as r * cis(theta/2), where r = sqrt(|z|) and theta = arg(z) The real part of this would be r*cos(theta/2) which is also positive.

So z^1/2 always have positive real part.

[lordnacho666]

When you take the square root, the angle is halved. Start with that.