Proof of Euler’s Formula

Next, we will present the initial proof. Recall from calculus that

We can now separate the series for (e):

Let’s denote (x) as an imaginary number, for example, (i(y)). Keep in mind that since (i) squared equals -1, then (i) to the fourth power is 1. Thus, we have

However, we can factor out (i) from the right series, leading us to:

This ultimately yields Euler’s formula when (y = 1):

Yay!

# Delving into Analysis

Differentiation

In this context, "analysis" broadly refers to calculus; if you are studying mathematics, you will encounter at least some aspect of analysis. Our previous findings indicate several ways to represent complex numbers:

I refer to these as the "standard," "vector," and "polar" forms of complex numbers. The terminology may vary slightly based on the source and its publication date.

Since (i) is a constant, we largely revert to standard calculus principles! Consider this:

For properly defined functions, as is customary. However, complications arise when dealing with a function expressed as (f(z)). Taking such a function where the output is complex leads us to:

Remember that for a multivariable derivative to be defined, limits must be consistent along any path, as this is integral to the derivative's definition in complex functions due to its limit nature.

Now, let’s simplify: for clarity, we assume the derivative along any path toward (z) is consistent (this aligns with the multivariable limit definition; if we can take limits concerning single complex numbers, we can similarly define such a derivative). We can isolate the real part:

Repeating this process for the imaginary axis (iy) yields (please verify this by substituting (iy)):

Definition 1 - Cauchy-Riemann Equations: These equations determine differentiability, allowing us to avoid calculating more complex difference quotients. They can be expressed more succinctly as:

Integration retains the same pleasant properties as it did with real numbers, including product, sum, and quotient rules. In many instances, L'Hopital's rule applies as well.

Integration

In the two-dimensional realm of our functions, we perform integrals along contours (curves traversing the complex plane). Here are a few key definitions:

• Smooth Arc: A smooth arc is defined as a one-to-one function given by

  with the following requirements:

  1. (z(t)) possesses a continuous derivative with respect to time.
  2. (z'(t)) is never zero (this ensures that the tangent vector does not have zero magnitude; otherwise, it disrupts the function's values. It also means no corners or cusps).

• Smooth Closed Curve: A smooth closed curve necessitates that the endpoints (a) and (b) possess the same differentiability properties, and that (z(b) = z(a)).

• Contour: A contour is a combination of smooth arcs that share endpoints, serving as a piecewise method to form a curve.

A non-smooth path can be transformed into a smooth one because it is composed of smooth arcs. This concept is what contours generalize. In this context, we will not delve into highly pathological examples, such as fractals.

• Contour Integral: A contour integral (sometimes referred to as a line integral) computes the area under the function sliced by your contour. We usually evaluate it using the chain rule:

Now, let’s consider integrating along a closed contour, such as the unit circle. For a parameterized path (z(t)), typically denoted as (C) or similar, this is expressed as:

Now, onto the core of the complex analysis experience: the Cauchy Integral formula. I will present this without proof, as the proof necessitates a solid understanding of certain complex functions, such as the logarithm.

Definition 2 - Cauchy Integral Formula: For a differentiable function (f) within a closed, bounded area defined by a contour (C), we find:

Hold on! This is quite a lot to digest. We observe that the values inside the region are entirely determined by the values along the boundary of the region. This resonates with Green’s Theorem or the Mean Value Theorem.

It also allows us to establish one of the most powerful theorems in mathematics:

Definition 3 - Cauchy Derivative Formula: By applying the first derivative concerning (w) to the previous example, we arrive at:

Extending this concept leads us to:

Incredible! We just discovered that if a function is differentiable at a point, it remains differentiable indefinitely!

# What Comes Next?

In the forthcoming article, I plan to explore additional significant outcomes from Complex Analysis, such as the Residue Theorem, introduce the Gamma function, and progress towards Analytic Number Theory. Ultimately, I will conclude with a discussion on fractional calculus, its connections to Complex Analysis, and potentially one more piece on Möbius geometry and Klein’s Erlanger Program.

What topics would you like me to address in the upcoming articles? Please share your thoughts below!