#euler's formula

The formula Maura was referring to is: e^(i * π) + 1 = 0

It’s a full celebration of five fundamental constants in one breathtaking equation. ^_^

Since it’s Pi-day (March 14), let’s appreciate Euler’s number e by defining π with it. 😎

e^(i * x) = cos(x) + i * sin(x)

Then we can define π as:

π = min { x ∈ ℝ⁺ | cos(x) = -1 and sin(x) = 0 }

Both parts are at least somewhat related to each other. I'll clarify for any FAQ's later when I have the spoons.

Euler’s Formula

When working with complex numbers, a formula that comes up very often is 𝑟 𝑒ᶿⁱ = 𝑟 (cos(𝜃) + sin(𝜃) 𝑖), sometimes seen as 𝑒ᶿⁱ = cos(𝜃) + 𝑖 sin(𝜃), known as Euler’s Formula after Swiss mathematician Leonhard Euler. This formula gives the relationship between the polar (𝑟 𝑒ᶿⁱ) and rectangular (𝑎 + 𝑏 𝑖) forms of complex numbers.

Wait, Complex Exponents?

When someone is first introduced to the polar form, a common reaction is to ask “what does it mean to raise a number to a complex power?” We generally think of exponents as denoting repeated multiplication: 6² = 6 × 6, 12⁵ = 12 × 12 × 12 × 12 × 12, and so on. Given 𝑏ⁿ, we multiply together 𝑛 copies of the base 𝑏. However, this definition of exponentiation only works for positive real integer exponents. Extending the definition of exponentiation to zero and negative integers is easy, and though non-integer values are much trickier, a smooth, continuous exponential function 𝑓(𝑥) = 𝑏ˣ is well-defined for positive, real bases and real number values of 𝑥.

This brings us to 𝑒ˣ, the exponential function with base 𝑒: a mathematical constant with a value of approximately 2.718, often known as Euler’s Number or the exponential constant. It is the only base such that the derivative of 𝑏ˣ is, itself, 𝑏ˣ, with a value and slope of 1 at 𝑥 = 0 and a value and slope of 𝑒 at 𝑥 = 1. It was first defined relating to the study of compound interest, as the limit of the expression (1 + 1 / 𝑛)ⁿ as 𝑛 goes to infinity. The function 𝑒ˣ is sometimes called the natural exponential function.

An Introduction to Taylor Series

Any continuous function can be approximated by an infinite sum (a.k.a. series) of terms in the form of an infinite polynomial. This infinite sum is called a Taylor Series, after English mathematician Brook Taylor. Truncating the infinite series after 𝑛 terms gives the 𝑛th Taylor Polynomial. Each finite polynomial is only an approximation, but the approximation becomes more accurate for every additional term that is added.

The Taylor Series of a function 𝑓(𝑥) at the input value 𝑎 is of the form 𝑓(𝑎) + 𝑓′(𝑎) (𝑥 − 𝑎) / 1! + 𝑓″(𝑎) (𝑥 − 𝑎)² / 2! + 𝑓‴(𝑎) (𝑥 − 𝑎)³ / 3! + … + 𝑓⁽ⁿ⁾(𝑎) (𝑥 − 𝑎)ⁿ/ 𝑛!, where:

𝑓⁽ⁿ⁾(𝑎) is the 𝑛th derivative of the function evaluated at 𝑎,

the zeroth derivative of a function is the function itself,

the zeroth power of a term is defined to be 1,

the exclamation mark denotes the factorial operation (multiplying every integer up to the indicated value, e.g. 3! = 1 × 2 × 3 = 6), and

zero factorial is defined to be 1.

Note that a Taylor Series can only be constructed for a function whose value and derivatives are known for the chosen input 𝑎, but — once constructed — can be used to calculate accurate approximations of the output values for other inputs close to 𝑎.

Let’s construct some Taylor Series to see how they work. The derivatives of the sine function sin(𝑥) cycle through four functions {sin(𝑥), cos(𝑥), −sin(𝑥), −cos(𝑥)}, and the values of the derivatives at 𝑥 = 0 cycle through the values {0, 1, 0, −1}. Thus, the Taylor Series for sin(𝑥) at 𝑎 = 0 is 𝑥 − 𝑥³ / 3! + 𝑥⁵ / 5! − 𝑥⁷ / 7! + … + (−1)ⁿ x²ⁿ⁺¹ / (2𝑛 + 1)!.

The derivatives and values of the cosine function cos(𝑥) at 𝑥 = 0 cycle through the same four functions and values as sine, offset by one. Thus, the Taylor Series for cos(𝑥) at 𝑎 = 0 is 1 − 𝑥² / 2! + 𝑥⁴ / 4! − 𝑥⁶ / 6! + … + (−1)ⁿ 𝑥²ⁿ / (2𝑛)!.

Finally, let’s construct the Taylor Series for 𝑒ˣ at 𝑎 = 0. Because the derivative of 𝑒ˣ is 𝑒ˣ, and the value of 𝑒⁰ is 1, the terms of the series simplify nicely, and we get 1 + 𝑥 + 𝑥² / 2! + 𝑥³ / 3! + … + 𝑥ⁿ / 𝑛!.

Putting it Together

Notice that the Taylor Series gives a definition of 𝑒ˣ that takes the variable 𝑥 out of the exponent. It is often denoted as exp(𝑥) to emphasize this fact. This provides a way to compute the output of the function 𝑒ˣ for values of 𝑥 that may not make sense as exponents themselves — like complex numbers. The function 𝑒ˣⁱ may not seem intuitive, but the function 1 + 𝑥 𝑖 + (𝑥 𝑖)² / 2! + (𝑥 𝑖)³ / 3! + … + (𝑥 𝑖)ⁿ / 𝑛! is easy to compute.

Because powers of 𝑖 cycle through four values {1, 𝑖, −1, −𝑖}, and because of the exponential identity (𝑏 𝑐)ⁿ = 𝑏ⁿ 𝑐ⁿ, we can expand the terms of the Taylor Series for 𝑒ˣⁱ as 1 + (𝑥 𝑖) / 1! − 𝑥² / 2! − (𝑥³ 𝑖) / 3! + 𝑥⁴ / 4! + (𝑥⁵ 𝑖) / 5! − 𝑥⁶ / 6! − (𝑥⁷ 𝑖) / 7! + … .

We can group the even and odd terms, and factor out the common 𝑖 from all odd terms, to get (1 − 𝑥² / 2! + 𝑥⁴ / 4! − 𝑥⁶ / 6! + …) + 𝑖 (𝑥 / 1! − 𝑥³ / 3! + 𝑥⁵ / 5! − 𝑥⁷ / 7! + …).

This gives us a sum of two separate Taylor Series, one with terms of the form (−1)ⁿ 𝑥²ⁿ / (2𝑛)! and the other with terms of the form (−1)ⁿ x²ⁿ⁺¹ / (2𝑛 + 1)!. The first is the Taylor Series for the function cos(𝑥), and the second is the Taylor Series for sin(𝑥) multiplied by 𝑖 — giving us 𝑒ˣⁱ = cos(𝑥) + 𝑖 sin(𝑥).

Dr. Michael Miller has announced his Autumn mathematics course, and it is…

Introduction to Complex Analysis

Course Description

Complex analysis is one of the most beautiful and useful disciplines of mathematics, with applications in engineering, physics, and astronomy, as well as other branches of mathematics. This introductory course reviews the basic algebra and geometry of complex numbers; develops the theory of complex differential and integral calculus; and concludes by discussing a number of elegant theorems, including many–the fundamental theorem of algebra is one example–that are consequences of Cauchy’s integral formula. Other topics include De Moivre’s theorem, Euler’s formula, Riemann surfaces, Cauchy-Riemann equations, harmonic functions, residues, and meromorphic functions. The course should appeal to those whose work involves the application of mathematics to engineering problems as well as individuals who are interested in how complex analysis helps explain the structure and behavior of the more familiar real number system and real-variable calculus.

Prerequisites

Basic calculus or familiarity with differentiation and integration of real-valued functions.

Details

MATH X 451.37 – 268651  Introduction to Complex Analysis Fall 2016 Time 7:00PM to 10:00PM Dates Tuesdays, Sep 20, 2016 to Dec 06, 2016 Contact Hours 33.00 Location: UCLA, Math Sciences Building Standard credit (3.9 units) $453.00 Instructor: Michael Miller Register Now at UCLA

  For many who will register, this certainly won’t be their first course with Dr. Miller — yes, he’s that good! But for the newcomers, I’ve written some thoughts and tips to help them more easily and quickly settle in and adjust: Dr. Michael Miller Math Class Hints and Tips | UCLA Extension

I often recommend people to join in Mike’s classes and more often hear the refrain: “I’ve been away from math too long”, or “I don’t have the prerequisites to even begin to think about taking that course.” For people in those categories, you’re in luck! If you’ve even had a soupcon of calculus, you’ll be able to keep up here. In fact, it was a similar class exactly a decade ago by Mike Miller that got me back into mathematics. (Happy 10th math anniversary to me!)

I look forward to seeing everyone in the Fall!

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A complex number z = a + bi has real part Re(z) = a and imaginary part Im(z) = b. Since a complex number has two independent parts, we may represent it on the xy-plane by plotting Re(z) on the x-axis and Im(z) on the y-axis; this xy-plane is called the complex plane,…

We’ve looked at how you can translate a sinusoidal voltage or current between trigonometric, polar, and rectangular coordinate forms.

Being able to do this makes it way easier to deal with these mathematically: rather than remembering a whole bunch of trig formulas and identities, we can just stick with regular addition, subtraction, multiplication, and division.

Euler’s formula is the reason we can do this. Euler’s formula is both simple and powerful and it comes up over and over again in electrical engineering. It relates the sine and cosine functions to both the imaginary number j (or i, if you’re not dealing with electrical engineering) and the constant e. It looks like this:

Looks like the way we did rectangular coordinates, right? That’s exactly what it is.

We don’t normally write polar notation with the exponential, but that’s just what it is. When we write the polar expression A /_ φ, it’s just a shorthand for Ae^(jφ). Knowing this, you can see why the translation between polar and rectangular coordinates makes sense:

What the hell?! There are two Euler's Formulas?!

Euler's Formula