Complex Analysis

Formulas

In many problems of real analysis—computing integrals such as ∫₋∞^∞ dx/(1+x²), solving differential equations, potential theory—the answer is surprisingly simple, but obtaining it “directly” is hard. It turns out that all these problems are transparently solved if we extend the real number line t...

A complex number z = x + iy is identified with the point (x, y) of the plane ℝ². Here x = Re z is the real part, y = Im z is the imaginary part, i is the imaginary unit (i² = −1).

Modulus: |z| = √(x² + y²) is the distance from the origin. Argument: arg z = arctg(y/x) is the angle with the positive semi-axis (defined up to 2πk). Conjugate number: z̄ = x − iy. Important: z·z̄ = |z|².

A function f: ℂ → ℂ is a mapping from the plane to itself: f(z) = u(x,y) + iv(x,y), where u and v are real-valued functions of two real variables. For example, f(z) = z² = (x+iy)² = x²−y² + 2ixy, so u = x²−y², v = 2xy.

Formulas

Many problems in mathematical physics are posed in complex domains (slits, airfoils, jagged surfaces). The method of conformal mappings allows one to transform such domains into simple regions (circle, half-plane), solve the problem there, and then map the solution back. This works because confor...

A holomorphic function $f$ with $f'(z_0) \neq 0$ is conformal at the point $z_0$: it preserves the angles between curves and their orientation.

Proof. Let $\gamma_1, \gamma_2$ be curves passing through $z_0$, $\gamma_1'(0) = v_1$, $\gamma_2'(0) = v_2$. Their images: $(f\circ\gamma_i)'(0) = f'(z_0)\cdot v_i$. Multiplication by $f'(z_0)\ne 0$ rotates both vectors by the same angle $\arg f'(z_0)$, so the angle between them is preserved. The...

This is a bijection of the extended plane $\mathbb{C}\cup\{\infty\}$ onto itself (i.e., a mapping of the Riemann sphere). Each Möbius transformation is a composition of translations, rotations, scalings, and inversion $z \mapsto 1/z$.

Formulas

Real differentiability is a “local” property. Holomorphicity, on the contrary, imposes extremely strict global constraints: knowing the function on a small segment, one can recover it everywhere; a bounded entire function must be constant. These theorems explain why complex numbers are algebraica...

Theorem (Liouville, 1847): Any holomorphic and bounded function on all of $\mathbb{C}$ is constant.

Proof via Cauchy’s inequality: $f'(z) = \frac{1}{2\pi i} \oint_{|\zeta - z| = R} \frac{f(\zeta)}{(\zeta - z)^2} d\zeta$. Estimate: $|f'(z)| \leq M/R \to 0$ as $R \to \infty$ ($M = \sup|f|$). Thus $f' \equiv 0$, hence $f = \mathrm{const}$.

This is a fundamental result: the functions $e^{\sin(z)}$ on $\mathbb{C}$ are unbounded because they “want” to be nontrivial.

Formulas

In real analysis, the integral over a closed contour of a conservative field is zero. In complex analysis, this same property—for holomorphic functions—generates something much more powerful: from the values of a function on the contour, one can exactly reconstruct its values at all interior poin...

Estimate: $|\int_\gamma f\,dz| \leq \max_{z \in \gamma} |f(z)| \cdot \text{length}(\gamma)$.

A complex integral decomposes as follows: $ \int_\gamma (u+iv)(dx + i\,dy) = \int_\gamma (u\,dx - v\,dy) + i \int_\gamma (v\,dx + u\,dy) $ — two real line integrals.

Theorem (Cauchy, 1825): If $f$ is holomorphic in a simply connected domain $D$ and $\gamma$ is a closed path in $D$:

Formulas

The Cauchy integral formula allows the computation of derivatives of all orders. From this, it immediately follows that a holomorphic function can be expanded as a convergent Taylor series—and this is an equality, not just an approximation. Holomorphy and analyticity in the complex case are the s...

Theorem: If $f$ is holomorphic in the disk $|z - z_0| < R$, then for all $z$ in this disk:

$ f(z) = \sum_{n=0}^{\infty} a_n (z - z_0)^n, \quad a_n = \frac{f^{(n)}(z_0)}{n!} = \frac{1}{2\pi i} \oint \frac{f(\zeta)}{(\zeta-z_0)^{n+1}} d\zeta. $

Radius of convergence $R = \mathrm{dist}(z_0, \partial D)$ — the distance to the nearest singular point. There is no “random” breakdown of convergence: the series converges precisely up to the nearest singularity.

Formulas

Holomorphic functions possess a remarkable property: knowing the function in a small domain allows one to extend it to everything permitted by analyticity. This is called analytic continuation—and it is unique. It is precisely this principle that allows the “definition” of the Riemann ζ-function ...

Theorem: If f and g are holomorphic in a connected domain D and coincide on a set with a limit point in D, then f ≡ g in D.

Corollary (analytic continuation principle): If f is holomorphic in D₁, g is holomorphic in D₂, D₁ ∩ D₂ ≠ ∅ is connected, and f = g on D₁ ∩ D₂, then g is the unique analytic continuation of f to D₂.

Physical meaning: There do not exist two different holomorphic functions that coincide even on a tiny segment. “A holomorphic object is an indivisible whole.”

Formulas

In the neighborhood of a point where a function is not defined or not holomorphic, the Taylor series is not applicable—it requires holomorphicity in a disk. The Laurent series expands this by including negative powers: $f(z) = \dots + c_{-2}/(z-z_0)^2 + c_{-1}/(z-z_0) + c_0 + c_1(z-z_0) + \dots$ ...

In an annulus $r < |z - z_0| < R$, a function $f$ that is holomorphic in the annulus is expanded as:

where the coefficients are: $c_n = \frac{1}{2\pi i} \oint_{|\zeta - z_0| = \rho} \frac{f(\zeta)}{(\zeta - z_0)^{n+1}} d\zeta$ ($r < \rho < R$).

Regular part: $\sum_{n=0}^\infty c_n (z - z_0)^n$—converges in the disk $|z - z_0| < R$. Principal part: $\sum_{n=1}^\infty c_{-n}/(z - z_0)^n$—converges outside the disk $|z - z_0| > r$.

Formulas

Many real integrals — from $\int_{-\infty}^{\infty} \frac{1}{1+x^2}dx$ to $\int_0^{\infty}\frac{\sin(x)}{x}\,dx$ — are difficult to compute using elementary methods. The method of residues allows one to reduce them to the calculation of a few numbers — the residues at the poles of the function. T...

Definition: The residue of $f$ at an isolated singular point $z_0$: $ \operatorname{Res}_{z=z_0} f = c_{-1} $ — the coefficient at $(z-z_0)^{-1}$ in the Laurent series.

Residue theorem: $ \oint_{\gamma} f(z) dz = 2\pi i \sum_k \operatorname{Res}_{z=z_k} f, $ where the summation is over all singular points $z_k$ inside the contour $\gamma$.

Simple pole: $\operatorname{Res}_{z=z_0} f = \lim_{z \to z_0} (z-z_0)\,f(z)$.

Formulas

The problem "how many zeros does a given function have in a given domain?" arises everywhere: in control theory (stability — zeros of the denominator in the right half-plane), in numerical methods (convergence of iterations), in cryptography (periods). The principle of the argument and Rouché's t...

Theorem: For a meromorphic function $f$ in a domain $D$, with $\gamma = \partial D$:

where $N$ is the number of zeros, and $P$ is the number of poles in $D$ (counted with multiplicities).

Geometric explanation: The integral equals the change in $\arg f$ when traversing $\gamma$, divided by $2\pi$; this is the winding number of the curve $f(\gamma)$ (the number of turns the image makes around the origin). Each zero contributes +1, each pole contributes −1.

Formulas

Not all integrals are reducible to trivial fractions. Integrals of the form $\int_0^\infty \ln(x)\cdot f(x) dx$, $\int_0^\infty x^a\cdot f(x) dx$, or sums $\sum f(n)$ require special contours and tricks. The two main methods are: the “keyhole" contour for power integrals and summation via residue...

For $\int_0^\infty x^a f(x) dx$ ($0 < a < 1$), we make a cut along $[0,+\infty)$ and integrate $f(z)\cdot z^a$ over the "keyhole" contour: a large circle $|z|=R \to$ disappears, a small $|z|=\varepsilon \to$ disappears, only the banks of the cut remain. On the upper bank $z = x$ $(x > 0)$, on the...

$\oint = \int_0^\infty x^a f(x) dx - e^{2\pi i a}\int_0^\infty x^a f(x) dx = (1 - e^{2\pi i a}) \int_0^\infty x^a f(x) dx = 2\pi i \sum \text{Res}$.

Key technique: $\pi\cot(\pi z)$ has simple poles at $z = n \in \mathbb{Z}$ with residue $1$. Then:

Formulas

The solution of many problems in mathematical physics — differential equations with variable coefficients — leads to functions that cannot be expressed in terms of polynomials, trigonometric functions, and exponentials. These are called *special functions*: the gamma function, Bessel functions, h...

Analytic continuation: $\Gamma(s)$ extends to $\mathbb{C} \setminus \{0, -1, -2, \ldots \}$ with simple poles at $s = 0, -1, -2, \ldots$ and residues $\operatorname{Res}_{s=-n} \Gamma = \frac{(-1)^n}{n!}$.

Functional equation: $\Gamma(s+1) = s\cdot\Gamma(s)$. Consequently: $\Gamma(n) = (n-1)!$ for natural $n$.

Reflection formula (Euler): $\Gamma(s)\cdot\Gamma(1-s) = \frac{\pi}{\sin(\pi s)}$.

Formulas

The Riemann mapping theorem guarantees the existence of a conformal mapping of any simply connected domain onto the unit disk. But how "large" can the distortion be? The Bieberbach theorem (Bieberbach conjecture, proved by de Branges in 1985) gives an exact answer in terms of the expansion coeffi...

Class S: Functions $f$ that are holomorphic and univalent (injective) in the unit disk $\mathbb{D}$, normalized by $f(0) = 0$, $f'(0) = 1$.

The Koebe Function: $k(z) = \frac{z}{(1-z)^2} = z + 2z^2 + 3z^3 + \ldots$ ($a_n = n$). This is the “extremal” function in class $S$: the image $k(\mathbb{D})$ is the whole plane minus the ray $(-\infty, -1/4]$.

Theorem (Koebe's $1/4$-theorem): The image $f(\mathbb{D})$ for $f \in S$ contains the disk $|w| < 1/4$. The estimate is sharp: for $k(z)$, the image $= \mathbb{C} \setminus (-\infty, -1/4]$.

Formulas

A polynomial $P(z)$ of degree $n$ has exactly $n$ zeros (with multiplicities) and can be decomposed as $P(z) = c(z-z_1)\cdots(z-z_n)$. Can we similarly "decompose" an entire function with infinitely many zeros? The Weierstrass theorem gives an affirmative answer, but with a caveat: to ensure the ...

An entire function is one that is holomorphic on all of $\mathbb{C}$. Examples: polynomials, $e^z$, $\sin z$, $\cos z$.

Order of growth: $\rho = \limsup_{r \to \infty} \frac{\ln \ln M(r)}{\ln r}$, where $M(r) = \max_{|z|=r} |f(z)|$.

To ensure the convergence of the infinite product, the canonical factor is introduced:

A rational function $R(z) = P(z)/Q(z)$ is decomposed into a sum of simple fractions: $R = \sum A_k/(z - z_k)^{m_k} +$ a polynomial. Mittag-Leffler's theorem is an analogue for meromorphic functions with infinitely many poles: one can "prescribe" the poles and their principal parts and construct a...

Theorem (Mittag-Leffler, 1877): Let $\{a_n\}$ be a sequence of points without an accumulation point in $\mathbb{C}$ (i.e., $|a_n| \to \infty$), and for each $a_n$ a "principal part" $p_n\left(1/(z - a_n)\right)$ is given—a polynomial in $1/(z - a_n)$. Then there exists a meromorphic function $f$ ...

$ f(z) = h(z) + \sum_n \left[ p_n \left( \frac{1}{z - a_n} \right) - q_n(z) \right], $

where $h(z)$ is an arbitrary entire function, and $q_n(z)$ are polynomials ensuring the convergence of the series.

Formulas

Differential equations are the bane of mathematical physics. The Laplace transform replaces them with algebraic equations: differentiation ↦ multiplication by s. By solving the algebraic equation and applying the inverse transform (via residues!), we obtain the solution to the differential equati...

Here σ₀ is the abscissa of convergence (determined by the growth of f as t → ∞).

where summation is over all poles of F (in the left half-plane for stable systems).

The Laplace transform is a Fourier transform "with exponential weighting": F(σ+iω) = ℱ[e^{−σt}f(t)](ω). The poles of F(s) correspond to the system's frequencies; stability ⟺ all poles in the left half-plane (Re s < 0).