Taylor and Laurent Series | Mathematics

Theorem 7.1. If $f$ is analytic on $|z - z_0| \lt R$ Then

$f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!}(z - z_0)^n$

And the series converges uniformly on compact subsets of $|z - z_0| \lt R$.

Proof. For $|z - z_0| \lt r \lt R$Apply Cauchy”s integral formula on $|\zeta - z_0| = r$:

$f(z) = \frac{1}{2\pi i}\int_{|\zeta - z_0| = r} \frac{f(\zeta)}{\zeta - z}\, d\zeta$

Write $\frac{1}{\zeta - z} = \frac{1}{(\zeta - z_0) - (z - z_0)} = \frac{1}{\zeta - z_0} \cdot \frac{1}{1 - (z - z_0)/(\zeta - z_0)}$ $= \sum_{n=0}^{\infty} \frac{(z - z_0)^n}{(\zeta - z_0)^{n+1}}$ (geometric series, convergent since $|z - z_0|/|\zeta - z_0| \lt 1$).

Substituting and integrating term by term gives the Taylor series. $\blacksquare$

Remark. The radius of convergence $R$ is the distance from $z_0$ to the nearest singularity of $f$.

$e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!} = 1 + z + \frac{z^2}{2!} + \cdots$

$\sin z = \sum_{n=0}^{\infty} \frac{(-1)^n z^{2n+1}}{(2n+1)!}$

$\cos z = \sum_{n=0}^{\infty} \frac{(-1)^n z^{2n}}{(2n)!}$

$\frac{1}{1 - z} = \sum_{n=0}^{\infty} z^n, \quad |z| \lt 1$

$\ln(1 + z) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} z^n}{n}, \quad |z| \lt 1$

Theorem 7.2 (Laurent Series). If $f$ is analytic on the annulus $r \lt |z - z_0| \lt R$ Then

$f(z) = \sum_{n=-\infty}^{\infty} a_n(z - z_0)^n = \cdots + \frac{a_{-2}}{(z - z_0)^2} + \frac{a_{-1}}{z - z_0} + a_0 + a_1(z - z_0) + \cdots$

Where

$a_n = \frac{1}{2\pi i}\int_\gamma \frac{f(z)}{(z - z_0)^{n+1}}\, dz$

For any simple closed contour $\gamma$ in the annulus encircling $z_0$.

The principal part is $\sum_{n=-\infty}^{-1} a_n(z - z_0)^n$ (negative powers). The analytic Part is $\sum_{n=0}^{\infty} a_n(z - z_0)^n$ (non-negative powers).

The Laurent series expansion depends on the annulus of convergence. A function may have different Laurent expansions in different annuli.

Proposition 7.3. The Laurent series expansion of $f$ in an annulus is unique.

• Wrong annulus choice. A function has different Laurent expansions in different annuli (e.g.\ $1/[z(z-1)]$ has distinct series in $0<|z|<1$ vs.\ $|z|>1$). Always identify all singularities and the correct annulus before expanding.

Definition. The residue at infinity of $f$ is defined as

$\mathrm{Res}(f, \infty) = -\frac{1}{2\pi i}\int_{|z|=R} f(z)\, dz$

For sufficiently large $R$ (enclosing all finite singularities).

Proposition 7.4. For a function $f$ with finitely many singularities in $\mathbb{C}$:

$\sum_{\mathrm{all\ finite\ } z_k} \mathrm{Res}(f, z_k) + \mathrm{Res}(f, \infty) = 0$

Proof. By the residue theorem applied to $|z| = R$ enclosing all finite singularities:

$\int_{|z|=R} f\, dz = 2\pi i \sum_{\mathrm{finite} \mathrm{Res}(f, z_k)}$.

But $\mathrm{Res}(f, \infty) = -\frac{1}{2\pi i}\int_{|z|=R} f\, dz$ So the sum is zero. $\blacksquare$

flowchart TD
    A[7_Taylor And Laurent Series] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

Every analytic function can be expanded as a power series, like a polynomial of infinite degree. This is possible because analyticity is so restrictive that the function is completely determined by its behavior at a single point. Taylor series capture the function near a point of analyticity. Laurent series extend this to functions with singularities, adding negative powers that encode the residue. The residue is like a local fingerprint of the singularity: it measures how much the function winds around that point, connecting local algebraic data to global topological information about contours.

• Cauchy’s Integral Formula: The integral formula provides the coefficients for Taylor and Laurent series expansions.

• Singularities and Residue Theory: Laurent series reveal the structure of singularities and enable residue computation.

• Analytic Continuation: Laurent series can extend functions beyond their original domain through analytic continuation.

• Classical Mechanics

• Electromagnetism

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Ensure you have mastered the prerequisite material before attempting this advanced content.

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Ensure you have mastered the prerequisite material before attempting this advanced content.