Glossary

Complex-differentiable at every point of some open set around z_0, not just at z_0 alone.

Extending an analytic function's domain by patching together overlapping power-series pieces.

Preserving the size of angles between curves, without necessarily preserving their orientation.

A ring-shaped region between two concentric circles, r<|z-z_0|<R.

A function F with F'(z)=f(z) throughout a domain -- lets a contour integral be evaluated from endpoints alone.

A point where every neighborhood catches both points inside and outside the set.

A set that fits inside some disk of finite radius; unbounded otherwise.

A single-valued piece cut out of a multivalued function by restricting its output range.

A curve removed from the domain to make a chosen branch continuous everywhere else.

A point around which a multivalued function cannot be split into consistent single-valued branches.

The contour integral \frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}F(s)e^{st}\,ds that recovers f(t) from its Laplace transform F(s).

P.V. \int_{-\infty}^\infty f(x)\,dx := \lim_{R\to\infty}\int_{-R}^R f(x)\,dx, a symmetric limit that can exist even when the ordinary improper integral does not.

u_x=v_y and u_y=-v_x -- the partial-derivative conditions a complex derivative forces on u and v.

A contour whose starting point and ending point are the same.

A set that contains every one of its boundary points.

The number multiplying a variable or a power, like the 3 in 3z^2.

f'(z_0)=\lim_{h\to 0} \dfrac{f(z_0+h)-f(z_0)}{h}, where h is a complex number approaching 0 from any direction.

A rule w=f(z) assigning a complex number to each complex number in some domain.

\log z = \ln|z| + i\arg z -- multivalued through \arg z, unless restricted to a branch.

An analytic function \Omega(z)=\varphi+i\psi packaging a 2D flow's velocity potential and stream function.

z^w := e^{w\log z}, generally multivalued since \log z is.

The real and imaginary parts of f(z), written as two-variable real functions u(x,y) and v(x,y).

A map that preserves both the size and the orientation of angles between curves.

A set that is all one piece -- it cannot be split into two separated nonempty parts.

The property of having no jumps or breaks: nearby inputs give nearby outputs.

f is continuous at z_0 when \lim_{z\to z_0} f(z) = f(z_0): no jump, hole, or blow-up there.

A piecewise-smooth curve, traced out in a definite direction, used as the path for a contour integral.

Sliding a contour to a more convenient one without changing a contour integral's value, as long as no singularity is crossed.

\int_C f(z)\,dz, the integral of f along a contour C, computed via \int_a^b f(z(t))z'(t)\,dt.

To settle toward one fixed value and get as close to it as you like.

A set where the straight segment joining any two of its points never leaves the set.

(z_1,z_2;z_3,z_4)=\dfrac{(z_1-z_3)(z_2-z_4)}{(z_1-z_4)(z_2-z_3)}, unchanged by any Möbius transformation.

A statement that fixes what a word means. It is agreed on, not proved.

Having a derivative at a point: the function has a well-defined rate of change there.

To take the derivative of a function.

Finding a harmonic function on a region that matches prescribed values on the boundary.

The set of points within a fixed distance of a center: |z-z_0|<r (open) or |z-z_0|\le r (closed).

A bijective conformal self-map of the open unit disk.

|z_1-z_2|, the ordinary plane distance between the points z_1 and z_2.

To split one number by another and get a quotient.

An open, connected set -- the standard setting for defining a complex function.

Analytic (holomorphic) at every point of the whole complex plane.

An isolated singularity that is neither removable nor a pole -- its principal part has infinitely many nonzero terms.

The small raised number that says how many times to multiply a base by itself.

The function e^z, which turns adding into multiplying and ties angles to rotations.

The complex plane together with the point at infinity, written \mathbb{C}\cup\{\infty\}.

A point z_0 with T(z_0)=z_0 -- a non-identity Möbius transformation has exactly one or two.

A point z_0 with f(z_0)=z_0, classified as attracting or repelling by the size of |f'(z_0)|.

A pair (analytic function, domain) -- one piece used to build up an analytic continuation.

An identity, such as \Gamma(z+1)=z\Gamma(z), relating a function's values at related inputs.

\Gamma(z)=\int_0^\infty t^{z-1}e^{-t}\,dt for \operatorname{Re}(z)>0, extending factorial to complex z.

A circle or a line -- the two kinds of set every Möbius transformation sends to another one.

All points on one side of a line, e.g. \operatorname{Re}(z)>0.

A harmonic function v paired with u so that u+iv is analytic.

A real function satisfying Laplace's equation, u_{xx}+u_{yy}=0.

Another name for analytic -- complex-differentiable throughout an open neighborhood of every point in a set.

A condition a theorem assumes. The conclusion is promised only when every hypothesis holds.

An equation that is true for every allowed input, unlike an equation you solve for particular ones.

The set of all output values f(z) as z ranges over a given set.

The number multiplying i in x+iy; it sets the height above the horizontal axis.

A contour with a small detour (usually a tiny semicircle) routed around a singularity that sits on it.

A whole number: positive, negative, or zero.

The function sitting inside an integral sign, the thing being integrated.

To find the total accumulated by a function along an interval or a path.

A point of a set with some whole neighborhood also inside the set.

The map w=1/z, which turns the plane inside out about the unit circle.

A point where f fails to be analytic, but f is analytic on some punctured disk around it.

The boundary between points that stay bounded and points that escape under iterating f.

A big circle with a thin slit cut along a branch cut and a small circle routed around the branch point, joined into one closed contour.

u_{xx}+u_{yy}=0 -- the defining equation of a harmonic function.

A series \sum_{n=-\infty}^\infty a_n(z-z_0)^n with both positive and negative powers, representing f on an annulus.

\lim_{z\to z_0} f(z)=L: f(z) gets arbitrarily close to L as z approaches z_0 from any direction at all.

The alternate name for a Möbius transformation, w=\dfrac{az+b}{cz+d}.

A map w=az+b (a\neq 0) built from a rotation, a scaling, and a translation of the plane.

The set of all points satisfying a stated geometric condition.

The set of c for which the orbit of 0 under z\mapsto z^2+c stays bounded.

Analytic on a domain except for isolated poles -- no other kind of singularity is allowed.

\left|\int_C f(z)\,dz\right|\le ML: the integral is bounded by a bound M on |f| times the contour's length L.

The distance of a complex number from the origin, written |z|. Also called its absolute value or size.

The (informal) fact that continuing a function around a loop back to its start can return a different value.

A domain with one or more holes -- some closed loop inside it cannot be shrunk to a point without leaving.

A rule assigning more than one output to some input, like the complex argument or logarithm.

A map w=\dfrac{az+b}{cz+d} with ad-bc\neq 0, also called a linear fractional transformation.

A boundary curve past which a power series admits no analytic continuation whatsoever.

Any open disk around a point, used as the standard notion of "nearby."

Finding a harmonic function on a region matching prescribed normal-derivative values on the boundary.

Not always the same value; the function changes somewhere.

A set in which every point has some neighborhood entirely inside the set.

Describing a curve with one running variable t, so each value of t gives one point.

A contour integral whose value depends only on the endpoints, not on which contour joins them.

A curve made of finitely many smooth pieces joined end to end, with at most finitely many corners.

A single extra point, denoted \infty, added to the complex plane to describe where z goes as |z|\to\infty.

The weighting function that reconstructs a harmonic function on a disk from its boundary values.

An isolated singularity where f(z)\to\infty; its order is the highest negative power in the Laurent series' principal part.

A closed flat shape made of straight sides, like a triangle or a rectangle.

A set where any two points can be joined by finitely many straight segments that stay inside it.

A sum of whole-number powers of z with fixed coefficients, like z^3-2z+5.

Traversed counterclockwise, so the region a closed contour encloses stays on the left of the direction of travel.

An infinite series \sum_{n=0}^\infty a_n(z-z_0)^n in powers of (z-z_0), with fixed coefficients a_n.

The one argument value in (-\pi,\pi], written \operatorname{Arg} z, singled out from the infinitely many valid arguments of z.

The standard, by-convention choice of branch, usually cut along the negative real axis.

The negative-power half of a Laurent series, \sum_{n=1}^\infty a_{-n}(z-z_0)^{-n}.

The one output a principal branch reports, singled out by convention from a multivalued function's full set of answers.

A disk with its center removed: 0<|z-z_0|<r.

The radius R of the largest disk around a power series' center on which it converges.

A polynomial divided by a polynomial, like \frac{z+1}{z^2-4}.

Flipping every point across a line, like a mirror image.

An informal name for a subset of the complex plane, before any sharper label is attached.

An isolated singularity where f extends to be analytic there too, once assigned the right value.

\operatorname{Res}_{z=z_0} f = a_{-1}, the coefficient of (z-z_0)^{-1} in the Laurent series at z_0.

\operatorname{Res}_{z=\infty} f := -\operatorname{Res}_{w=0}\left[\frac{1}{w^2}f\!\left(\frac1w\right)\right], chosen so all residues, including infinity's, sum to zero.

The unit sphere used to picture the extended complex plane, with the north pole standing for \infty.

Turning every point around the origin by a fixed angle. Multiplying by e^{i\theta} does this.

A formula mapping the upper half-plane conformally onto the interior of a given polygon.

Half of a circle, cut along a diameter.

A contour that does not cross itself, except that a closed contour's two endpoints may coincide.

A pole of order 1 -- the Laurent series' principal part is a single term, a_{-1}(z-z_0)^{-1}.

A domain with no holes -- informally, every loop drawn inside it can be shrunk to a point without leaving the domain.

A value that makes an equation true when you put it in.

The map pairing each plane point with the sphere point on the line from it to the north pole.

The imaginary part \psi of a complex potential; its level curves are the flow's streamlines.

A curve a fluid particle actually travels along, given by a level curve \psi=\text{const}.

Replacing a variable with another expression to simplify a problem or change how it looks.

The power series \sum_{n=0}^\infty \frac{f^{(n)}(z_0)}{n!}(z-z_0)^n representing an analytic f near z_0.

The plane whose points are complex numbers, x+iy plotted as (x,y).

A statement that has been proved, so you can use it without proving it again.

The rule |z+w|\le|z|+|w|: one side of a triangle is never longer than the other two together.

A series of functions converges uniformly on a set when one tail bound works for every point of the set at once.

A corner of a polygon, where two sides meet.

The integer n(C,z_0)=\frac{1}{2\pi i}\oint_C \frac{dz}{z-z_0} counting how many times C loops around z_0.

A point z_0 with f(z_0)=0; its order is the index of the first nonvanishing derivative at z_0.