Suppose z=a+b i, w=u+i v, and a=((|w|+u)/(2))^1 / 2, b=((|w|-u)/(2))^1 / 2 Prove that z^2=w if

step 1 So, Z1 and Z2 are two complex numbers which satisfies this inequality, mod Z1 is less than 1 les

Suppose z=a+b i, w=u+i v, and a=((|w|+u)/(2))^1 / 2, ...

Suppose $z=a+b i, w=u+i v$, and
$$a=\left(\frac{|w|+u}{2}\right)^{1 / 2}, \quad b=\left(\frac{|w|-u}{2}\right)^{1 / 2}$$
Prove that $z^{2}=w$ if $v \geq 0$ and that $(z)^{2}=w$ if $v \leq 0 .$ Conclude that every complex number (with one exception!) has two complex square roots.

Key Concepts

Branch Selection and Principal Value

When working with multi-valued functions such as the square root, it is essential to choose a branch – a continuous determination of the function's value – to maintain consistency. The principal square root is often selected using conditions on the argument (such as nonnegative imaginary part) to provide a unique representative, with the other square root being its negative.

Square Roots in the Complex Plane

Every nonzero complex number has two distinct square roots, reflecting the fundamental property of quadratic equations having two solutions. This is analogous to the real number case but with added complexity due to the presence of an imaginary component. The process of finding these square roots often involves expressing the complex number in polar form and applying De Moivre’s theorem.

Modulus of a Complex Number

The modulus of a complex number is a measure of its distance from the origin in the complex plane. It is defined as |w| = ?(u² + v²) for a complex number w = u + iv, and it plays a crucial role when converting complex numbers between rectangular and polar forms as well as in operations like finding square roots.

Complex Numbers

Complex numbers extend the real numbers by incorporating an imaginary unit i, satisfying i² = -1. They are expressed in the form x + yi, where x and y are real numbers, allowing the representation of both magnitude and directional (phase) components in the complex plane.

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