Square Root of Complex Number

Complex numbers consist of two parts, namely a real number and an imaginary part. The number on the number line is the real number, and the imaginary unit is the multiple of i, where i = √-1.

The general form of the complex number is z = a + bi,

where

• a represent the real part,
• b represent the imaginary part, and
• i is the imaginary unit

 i² = -1. 3 + 2i, -5 - 4i are examples of complex numbers.

The square root of a complex number is another complex number that, when squared, gives the original complex number. If the square root of a complex number is √(a + ib) = ±(x + iy), then (x + iy)² = a + ib.
 

The simplest way to find the square root of a complex number in the form of x + iy, square it, and then compare the real and imaginary parts to determine the values of x and y. 

• Rectangular Form of a Complex Number:
  
  The rectangular form of a complex number shows it using Cartesian coordinates on the complex plane. It is written as a + bi, where a is the real part and b is the imaginary part. 

• Polar Form of a Complex Number:
  
  The polar form represents a complex number based on its distance from the origin and the angle it makes with the positive real axis. It is written as r(cosθ + isinθ), where r is the modulus of the complex number, and θ is the argument of the complex number. 

• Formulas for Conversion:

Rectangular to Polar Form:
Modulus (r) = √(a2 + b2)
Argument (θ) = tan-1ba

Polar to rectangular Form:
Real part (a) = r × cosθ
Imaginary part (b) = r × sinθ

The Square root of a complex number is found in a pair, similar to the square root of real numbers. When either of these values is squared, the result is the original complex number.

The general representation of the square root of (x + iy) is:

\(√(x + iy) = ±[√(√(x2 + y2) + √x2) + i.y/|y|. √(√(x2 + y2) - √x2)] \ or \ \\ √(x + iy) = ±[√(|z| + √x2) + i.y/|y|. √(|z| - √x2)]\)

Here, z = x + iy and y ≠ 0

A complex number can be written in polar form as:
z = r(cosθ + isinθ)

Where

• r is the modulus (distance from the origin),
• Θ is the argument (angle with the positive real axis).

To find the nth root of z, we use:

\(z1/2 = r1/ncos θ + 3600kn + isin θ + 3600k n\\ \ Or\ \\ z1/2 = r1/ncos θ + 2kn + isin θ + 2k n\)

Here, k = 0, 1, 2, …, n - 1, which means n has different roots. 

For the square root (when n = 2), the formula becomes:

\(z1/2 = r1/2cos θ + 3600k2 + isin θ + 3600k 2\\ \text{Or in radians:} \\ z1/2 = r1/2cos θ + 2k2 + isin θ + 2k 2\)

Where k = 0 or 1 because a square root always has two values.

If a complex number is written in polar form as:
z = r(cosθ + isinθ)

Then the n-th roots of z are given by:

\(z^k = r^{1/n} [cos ({θ + 2\pi k \over n}) + isin({ θ + 2k \over n})]\)

Here,
 

• r is the modulus of the complex number, found by r = |z|.
• θ is the angle that the number makes with the real axis.
• n is the type you want to find.
• k is a whole number starting from 0 up to n - 1.
• Each value of k gives a different root. 

Finding the square root of a complex number z = a + ib can be done in two easy ways: using the direct formula or using the polar form. Given below are the tips to remember. 

1. Using the Direct Formula:

The direct formula is quicker if we already know the values of a and b.
Write z = a + ib, where a is the real part and b is the imaginary part.

The modulus is |z| = a² + b²

Use the formula:
a + ib = |z| + a²  + i b|b||z| - a² 
 

• If b > 0, both x and y have the same sign.
• If b < 0, x and y have opposite signs.

2. Using Polar Form:

Polar form is better when you know the modulus and angle, or want to see the geometric meaning.

Convert z = a + ib into a polar form:
z = r(cosθ + isinθ)
where, r = |z| and θ = tan-1(ba)

Use the formula: 
\(z^{1/2} = r^{1/2}[cos ((θ + 2k)/ 2) + isin ((θ + 2k)/ 2)]\)

With k = 0, 1, to get the two roots.

The concept of square roots of complex numbers is widely used in various scientific and engineering fields. It often appears when solving differential equations, calculating impedances in electrical circuits, or when analyzing wave functions. Given below are some of the real-life applications and how they are used.

1. Signal Processing: In signal processing, complex numbers are fundamental for representing signals in the frequency domain. During operations such as the Fourier transform, square roots of complex numbers frequently arise when calculating the magnitude of frequency components, determining power, or adjusting phase. 
   
    
2. Communication Systems: Square roots are used in computing the magnitude of these signals, which is essential in determining power, signal strength, and error correction. Complex numbers represent modulated signals in telecommunication. 
   
    
3. Computer Graphics and 3D modeling: When doing certain geometric transformations or generating fractals, the square root of complex numbers is part of the calculations. Rotations, wave patterns, and wave phenomena are often modeled using complex numbers because they can represent both magnitude and phase in a single expression. 
   
    
4. Aerodynamics: In analyzing airflow over wings or fluid movement, complex potential functions are used to simplify real-world fluid equations. The square root of complex numbers appears when calculating velocity potentials or solving flow equations around objects.
   
    
5. Quantum Mechanics: Complex numbers represent the quantum states and wave functions. The probability amplitudes in quantum mechanics involve square roots when calculating probabilities or normalizing wave functions.