Exponential Form (Euler’s Relation) Revision Content

Basics of Exponential Form/Euler’s Relation

Exponential form is another way to represent complex numbers, alternative to the usual rectangular form (real + imaginary).
Euler’s relation is the basis for the exponential form of complex numbers. It links the fields of trigonometry and exponentiation.
Euler’s Formula states that for any real number x, e^(ix) = cos(x) + i sin(x). This relation uniquely captures the behaviour of both exponential and trigonometric functions.
A complex number can therefore be written in exponential form as: r*e^(iθ), where r is the modulus of the number (the magnitude) and θ is the argument (the angle).

Converting Between Forms

To convert from rectangular form (a + bi) to exponential form, you first calculate the modulus, r, using the Pythagorean Theorem (r = sqrt(a^2 + b^2)) and then find the argument, θ, using θ = atan2(b,a), where atan2 is a two-argument version of arctangent that preserves the quadrant of the point. Remember, atan2 values range from -π to π.
To convert from exponential form back to rectangular form, use Euler’s relation, replacing x in the formula with the argument θ.

Euler’s Identity

Euler’s Identity is a special case of Euler’s relation when x = π. The Identity is stated as e^(iπ) + 1 = 0. This is often hailed as a remarkable relationship connecting the fundamental numbers i, e, π, 1, and 0.

Operations in Exponential Form

Multiplication and Division: These operations are easier to perform in exponential form than in rectangular form. For multiplication, you multiply the moduli and add the arguments. For division, you divide the moduli and subtract the arguments.
Powers and Roots: Calculations involving powers and roots of complex numbers become straightforward in exponential form, thanks to De Moivre’s Theorem, which states that [r*e^(iθ)]^n = r^n * e^(inθ). Remember, when calculating roots, a complex number has not one but ‘n’ distinct roots.

Complex Logarithms

The natural logarithm, being the inverse of the exponential function, can also be derived for complex numbers using exponential form.
The principal value of a complex logarithm is defined as that corresponding to the principal value of the argument (from -π to π).

Polar Coordinates and Exponential Form

Because a complex number in exponential form can be visualised as a point in the plane with polar coordinates (r, θ), it makes it useful in fields involving geometry, physics, and engineering.

Please remember these formulas are fundamental to understanding other mathematical constructs. Regular practice is vital to good understanding and problem-solving skills.