Complex Analysis · aiwiki.page

Complex analysis is the branch of mathematical analysis concerned with functions of complex numbers. Its central objects are holomorphic functions: functions that possess a complex derivative throughout an open region. Although it extends familiar ideas from calculus, complex differentiability imposes unusually strong restrictions. Holomorphic functions have derivatives of every order, admit local power-series representations, and satisfy integral identities linking their interior values to values on surrounding curves. The subject combines analytical methods with geometric interpretations. (ocw.mit.edu)

Complex differentiability

A complex variable is written $z=x+iy$ , where $x,y$ are real numbers and $i^2=-1$ . A complex-valued function can therefore be expressed as

f(z)=u(x,y)+iv(x,y).

Its derivative at $z_0$ is

f'(z_0)=\lim_{h\to0}\frac{f(z_0+h)-f(z_0)}{h},

provided the limit exists independently of how the complex increment $h$ approaches zero. Unlike a real variable, $h$ can approach along any direction in the plane. A function is holomorphic on an open set if this derivative exists at every point of that set. (ocw.mit.edu)

Complex differentiability implies the Cauchy–Riemann equations,

u_x=v_y,\qquad u_y=-v_x.

Conversely, these equations imply holomorphy when the first partial derivatives are continuous throughout an open region. The equations express compatibility between the real and imaginary components, not merely separate differentiability. For example, $f(z)=z^2$ is holomorphic everywhere, whereas complex conjugation $f(z)=\bar z$ , despite being smooth as a mapping of two real coordinates, is nowhere complex differentiable. (ocw.mit.edu)

Contour integration and Cauchy’s theorems

Integration is performed along oriented curves in the complex plane. For a piecewise continuously differentiable curve $\gamma:[a,b]\to\mathbb C$ , the corresponding line integral is

\int_\gamma f(z)\,dz =\int_a^b f(\gamma(t))\gamma'(t)\,dt.

The Cauchy integral theorem states that this integral is zero around every closed such curve in a simply connected domain when $f$ is holomorphic there. The domain’s topology matters: $1/z$ is holomorphic away from zero, but its integral around a counterclockwise circle surrounding zero is $2\pi i$ . (warwick.ac.uk)

The Cauchy integral formula provides a stronger connection between integration and differentiation. If $f$ is holomorphic on a neighborhood of a closed disk, $\gamma$ is its counterclockwise boundary, and $z$ lies inside, then

f(z)=\frac{1}{2\pi i}\int_\gamma \frac{f(\zeta)}{\zeta-z}\,d\zeta.

Similar formulas recover every derivative. Thus boundary values determine the function throughout the disk, and the existence of one complex derivative on an open set forces derivatives of all orders. (ocw.mit.edu)

Series representations and rigidity

Every holomorphic function has a convergent Taylor series in a sufficiently small disk around each point:

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

Consequently, holomorphy and local representability by a convergent power series are equivalent. The convergence radius reflects how far that representation extends before encountering an obstruction such as a singularity. (ocw.mit.edu)

Several major theorems describe the resulting rigidity. The identity theorem states that two holomorphic functions on a connected domain coincide everywhere if they agree on a set with an accumulation point inside that domain. The maximum modulus principle prevents a nonconstant holomorphic function from attaining a local maximum of its absolute value in the interior. Liouville’s theorem states that every bounded entire function—one holomorphic on the whole plane—is constant. This yields a proof of the fundamental theorem of algebra, which guarantees a complex root for every nonconstant polynomial. (warwick.ac.uk)

Singularities and residues

Near an isolated singularity $a$ , a function holomorphic on a punctured disk has a Laurent series,

f(z)=\sum_{n=-\infty}^{\infty}c_n(z-a)^n.

The negative-power terms classify the singularity. With none, it is removable; with finitely many and at least one nonzero, it is a pole; with infinitely many nonzero negative-power terms, it is essential. Thus $\sin z/z$ has a removable singularity at zero, $1/z^2$ has a pole, and $e^{1/z}$ has an essential singularity there. (ocw.mit.edu)

The coefficient $c_{-1}$ is the residue. The residue theorem states that, for a positively oriented simple closed contour enclosing finitely many isolated singularities and no other failures of holomorphy,

\int_\gamma f(z)\,dz =2\pi i\sum_k\operatorname{Res}(f,a_k).

This reduces a global integral to local coefficients. More general contours introduce winding-number factors. Residue methods also evaluate many real definite and improper integrals. (ocw.mit.edu)

Geometry, continuation, and applications

A holomorphic function with nonzero derivative is locally a conformal map: it preserves oriented angles while changing lengths by a local scale factor. The Riemann mapping theorem states that every nonempty simply connected proper domain in the complex plane is conformally equivalent to the unit disk. Such transformations allow problems on complicated planar regions to be transferred to simpler ones. (ocw.mit.edu)

Analytic continuation extends a holomorphic function beyond its original domain by matching local representations. Complex logarithms illustrate why extensions can require branches: continuation around zero changes a logarithm’s value by $2\pi i$ . Continuation is also central to the study of special functions, including the gamma function. (ocw.mit.edu)

The real and imaginary parts of holomorphic functions are harmonic functions, satisfying Laplace’s equation. This connection supports applications to two-dimensional potential flow and boundary-value problems. In number theory, complex-variable methods applied to the Riemann zeta function underpin proofs of the prime number theorem. (ocw.mit.edu)