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If you've learned about Taylor series in calculus, you might know that some functions can be written as an infinite sum of powers of x x or z z . But what happens when a function has a problem—a singularity? That’s where the Laurent series comes in! What Is a Laurent Series? A Laurent series is like a Taylor series, but it includes negative powers too! It looks like this: f ( z ) = ∑ n = − ∞ ∞ c n ( z − z 0 ) n f(z) = \sum_{n=-\infty}^{\infty} c_n (z - z_0)^n Here: z 0 z_0 is the center of expansion, The coefficients c n c_n tell us how much each term contributes, The powers of ( z − z 0 ) (z - z_0) can be both positive and negative ! The key difference from the Taylor series is that Laurent series allow for terms like 1 ( z − z 0 ) k \frac{1}{(z - z_0)^k} , which help describe functions with singularities. Why Do We Need Laurent Series? Sometimes, a function behaves badly at a certain point. For example, the function: f ( z ) = 1 z f(z) = \frac{1}{z} is not define...

Introduction Have you ever wondered how mathematicians work with functions that "blow up" at certain points? In complex analysis , we use Laurent series to express functions with singularities (points where they become infinite or undefined). From these series, we can extract something called the residue , which plays a crucial role in evaluating contour integrals. In this post, we’ll learn how to compute residues using Laurent series by working through two examples: f ( z ) = e 1 / z f(z) = e^{1/z} , a function with an essential singularity at z = 0 z = 0 . g ( z ) = ln ⁡ ( 1 + z ) 1 + z 2 g(z) = \frac{\ln(1+z)}{1+z^2} , a function with isolated singularities that we will analyze. No prior knowledge of complex analysis is needed—just a curiosity for math! What is a Laurent Series? A Laurent series is like a Taylor series but allows for negative powers of z z . It takes the form: f ( z ) = ∑ n = − ∞ ∞ c n ( z − a ) n . f(z) = \sum_{n=-\infty}^{\infty} c_n (z-a)^n....

Introduction If you've ever worked with calculus, you know that limits, derivatives, and integrals are powerful tools for solving problems. But what if we told you that complex numbers can make certain problems even easier to solve? One of the key ideas in complex analysis is the residue of a function. Residues help us evaluate difficult integrals and understand the behavior of functions near their singularities (places where they "blow up"). Don't worry if you've never studied complex numbers beyond i = − 1 i = \sqrt{-1} . We'll take it step by step and use examples to show you how to compute residues. What is a Residue? The residue of a function at a singularity is the coefficient of 1 z − a \frac{1}{z-a} in its Laurent series expansion. In simpler terms, it's a special number that tells us how a function behaves near a problematic point. Residues are particularly useful when we use the Residue Theorem , which helps evaluate contour integrals (a...

Introduction Imagine you're navigating a maze, and every turn you take preserves the exact angles of the walls, no matter how complex the twists. This idea of preserving angles is at the heart of many important functions in mathematics and physics, and it’s exactly what the Cauchy-Riemann equations help us understand in complex analysis. In this post, we’ll explore: ✅ What the Cauchy-Riemann equations are. ✅ How they define differentiability for complex functions. ✅ Their derivation from first principles . ✅ How they’re applied in physics and engineering . No prior knowledge of complex analysis is required! If you've worked with derivatives in basic calculus , you're all set. 🚀 What Are the Cauchy-Riemann Equations? In real-number calculus, a function is differentiable if its derivative exists. But in complex numbers , differentiability is way more powerful than in real functions. A complex function must satisfy extra conditions to be differentiable, and t...

Introduction Have you ever wondered why some numbers are called deficient ? A number is deficient if the sum of its proper divisors (all its divisors except itself) is less than the number itself. But what happens when we take a prime number and raise it to a power? It turns out that every power of a prime number is always deficient! Today, we’ll explore: ✅ What deficient numbers are. ✅ How to calculate the sum of divisors of a number. ✅ A simple proof that all prime powers are deficient. By the end of this post, you’ll have a deeper understanding of a cool mathematical property of prime numbers! Let’s dive in. 🚀 What Are Deficient Numbers? A number N N is deficient if the sum of its proper divisors is less than the number itself. Mathematically, this means that if σ ( N ) \sigma(N) represents the sum of all divisors of N N , then: σ ( N ) − N < N \sigma(N) - N < N which simplifies to: σ ( N ) < 2 N \sigma(N) < 2N where σ ( N ) \sigma(N) includes ...