# Pictures of Julia and Mandelbrot Sets/Terminology

All the definitions are for points, functions, subsets, ... of the plane. We identify the points of the plane with the complex numbers and with vectors.

accumulation point (or cluster, or limit point) for a set: a point z such that each neighbourhood of z contains points of the set

boundary of a set: the set of points that are point of accumulation for the set as well as for the complement of the set

Cauchy-Riemann equations for a differentiable function ${\displaystyle f(x,y)=f_{x}(x,y)+if_{y}(x,y)}$ of the plane into itself: the two equations

${\displaystyle {\frac {\partial }{\partial {x}}}f_{x}={\frac {\partial }{\partial {y}}}f_{y}}$ and ${\displaystyle {\frac {\partial }{\partial {y}}}f_{x}=-{\frac {\partial }{\partial {x}}}f_{y}}$,

if they are satisfied, the function is differentiable as a complex function

cardioid: a heart-shaped curve (generated by a fixed point on a circle as it rolls round another circle of equal radius)

closed set: a set whose complement is an open set

complex number: a "two-dimensional" number, a number of the form (x, y), where x and y are real numbers, such a pair is usually written x+iy, where x and y are separated by the imaginary unit i, satisfying ${\displaystyle i^{2}=-1}$

convergence: a sequence ${\displaystyle z_{i}}$ (i = 0, 1, 2, ...) converges to the point z*, if for each neighbourhood U of z*, there exists a number N such that ${\displaystyle z_{i}}$ belongs to U for i > N. The sequence converges to the finite cycle C of order r, if for each point z* of the cycle the sequence ${\displaystyle z_{n+ir}}$ (for some n) converges to z*

countabel set: a set that can be put into a ono-to-one correspondance with the natural numbers, the rational numbers is a countabel set

critical point of a complex differentiable function f(z): a zero for the derivative f'(z)

cycle: a finite set of points in the plane, it can contain the point infinity, its number of elements is called its order

derivative (or differential quotient) of the complex function f(z) in the point z*: the number

${\displaystyle f'(z*)=(df(z)/dz)_{z=z*}=\lim _{h\to 0}(f(z*+h)-f(z*))/h}$

(h complex) if it exists

determinant of the 2x2 matrix {${\displaystyle a_{ij}}$}: the real number |{${\displaystyle a_{ij}}$}| = ${\displaystyle a_{11}a_{22}-a_{12}a_{21}}$

holomorphic (or analytic) function: a complex function defined on an open set of the plane, that is complex differentiable in every point of the open set, such a function possesses derivatives of all orders

interior point of a set: a point z such that a neighbourhood of z is contained in the set

iteration: repeated operation with the same function f(z): ${\displaystyle z_{1}=f(z_{0}),z_{2}=f(z_{1}),z_{3}=f(z_{2}),...}$

lim: if the sequence ${\displaystyle a_{n}}$ (n = 0, 1, 2, ...) converges to the number a, we write limn → ∞ an = a, if the value of the function f(z) converges to a for z converging to z*, we write limn → ∞ f(z) = a

matrix: a rectangular array of numbers {${\displaystyle a_{ij}}$} (i = 1, ..., m, j = 1, ..., n)

neighbourhood of a point z: a set containing a (small) circle with centre z

Newton iteration for an equation g(z) = 0: the iteration function ${\displaystyle f(z)=z-g(z)/g'(z)}$, if the sequence of iteration generated by a point converges to a fixed point, then this point is a solution to the equation g(z) = 0

norm of a complex number z = x+iy: the real number |z| = √(x2 + y2) ≥ 0

open set a set all points of which are interior points

partial derivative with respect to x of the function f(z) in the point z: the number

${\displaystyle \partial f(z)/\partial {x}=lim_{h\to 0}(f(z+h)-f(z))/h}$

where h is real (if it exists)

partial derivative with respect to y of the function f(z) in the point z: the number

${\displaystyle \partial f(z)/\partial {y}=lim_{h\to 0}(f(z+ih)-f(z))/h}$

where h is real (if it exists)

polynomial of degree n: a function of the form ${\displaystyle a_{0}+a_{1}z+a_{2}z^{2}+...+a_{n}z^{n}}$

rational function: a function of the form p(z)/q(z), where p(z) and q(z) are polynomials

scalar product of the two vectors ${\displaystyle v_{1}}$ = {${\displaystyle x_{1}}$, ${\displaystyle y_{1}}$} and ${\displaystyle v_{2}}$ = {${\displaystyle x_{2}}$, ${\displaystyle y_{2}}$}: the real number ${\displaystyle v_{1}*v_{2}=x_{1}x_{2}+y_{1}y_{2}=|v_{1}||v_{2}|cos(\theta )}$, where ${\displaystyle \theta }$ is the angle between ${\displaystyle v_{1}}$ and ${\displaystyle v_{2}}$

transcendental function: a function that cannot be constructed in a finite number of steps from elementary functions and their inverses, e.g. ${\displaystyle sin(z)=z-z^{3}/3!+z^{5}/5!-z^{7}/7!+...}$, where n! = 1x2x3x...xn

uncountabel set: a set that is not countable, such a set (belonging to the plane) can be put into a one-to-one correspondance with the real numbers

Rules for operation with complex numbers

i2 = -1
(x1 + iy1) + (x2 + iy2) = (x1 + x2) + i(y1 + y2)
(x1 + iy1)(x2 + iy2) = (x1x2 - y1y2) + i(x1y2 + x2y1)
(x1 + iy1)/(x2 + iy2) = ((x1x2 + y1y2) + i(-x1y2 + x2y1))/ (x22 + y22)

The conjugate number to ${\displaystyle z=x+iy}$ is ${\displaystyle z^{-}=x-iy}$, that is, the reflection of z in the x-axis. The norm of z is the real number |z| = √(x2 + y2). We have ${\displaystyle |z|^{2}=zz^{-}}$. From this, we can derive the rule for division: ${\displaystyle z/w=zw^{-}/(ww^{-})=zw^{-}/|w|^{2}}$.

A complex number z = x + iy can be written

${\displaystyle z=r(sin(\theta )+icos(\theta ))}$

where r is the norm of z:

r = |z| = √(x2 + y2)

and where the angle ${\displaystyle \theta }$ is the argument arg(z) of z:

${\displaystyle \theta }$ = arctan(y/x) for x > 0
${\displaystyle \theta }$ = arctan(y/x) + ${\displaystyle \pi }$ for x < 0

arg(z) is multivalued: arg(z) = ${\displaystyle \theta +n2\pi i}$, for every integer n.

The point ${\displaystyle sin(\theta )+icos(\theta )}$ (lying on the unit circle) is also denoted ${\displaystyle e^{i\theta }}$, and we define the exponential function ${\displaystyle exp(z)}$ by

${\displaystyle exp(z)=e^{z}=e^{x+iy}=e^{x}e^{iy}}$

The sinus and cosinus relations can be written

${\displaystyle cos(\theta _{1}+\theta _{2})+isin(\theta _{1}+\theta _{2})=(cos\theta _{1}+isin\theta _{1})(cos\theta _{2}+isin\theta _{2})}$

and from this we see that ${\displaystyle e^{z}}$ has the exponential property for z imaginary:

${\displaystyle e^{i(\theta _{1}+\theta _{2})}=e^{i\theta _{1}}e^{i\theta _{2}}}$

The logarithm function log(z) is defined as the inverse function to exp(z): log(z) = w if exp(w) = z. We have log(z) = log|z| + i arg(z). log(z) is multivalued: ${\displaystyle log(z)=w+n2\pi i}$, for every integer n.

For a positive real number a and a complex number z, we define ${\displaystyle a^{z}=e^{log(a)z}}$. For a complex number z, the power ${\displaystyle z^{n}}$ is only defined when the exponent n is an integer.

Differentiable complex function

A complex function defined on an open domain of the plane is called holomorphic (or analytic), if it is differentiable in every point of this domain. If that is so, the derived function f'(z) is also differentiable in every point (this theorem is not true for real functions). We have the usual rules for differentation:

${\displaystyle d(f(z)+g(z))/dz=df(z)/dz+dg(z)/dz}$

${\displaystyle d(f(z)g(z))/dz=(df(z)/dz)g(z)+f(z)(dg(z)/dz)}$

${\displaystyle (d(f(g(z))/dz)_{z*}=(df(z)/dz)_{g(z*)}(dg(z)/dz)_{z*}}$

${\displaystyle dz^{n}/dz=nz^{n-1}}$ (n integer)

${\displaystyle da^{z}/dz=log(a)a^{z}}$ (a positive real number)

${\displaystyle d(exp(z))/dz=exp(z)}$

${\displaystyle d(log(z))/dz=1/z}$

From these rules, we can derive all we need, for instance:

${\displaystyle d(f(z)^{-1})/dz=-f'(z)/f(z)^{2}}$

${\displaystyle d(log(f(z)))/dz=f'(z)/f(z)}$

For the computer it is easy to find f'(z): f'(z) = (f(z+h) - f(z))/h for ${\displaystyle h=10^{-9}}$, for instance.

The derivative of a function into the real numbers

The real function f(z) on a domain of the plane, is differentiable in the point z, if

limt → 0 (f(z + th) - f(z))/t

(t real) exists for every complex number h, and if the hereby defined function ${\displaystyle Df(z)(h)}$ from the complex numbers (h) into the real numbers satisfies ${\displaystyle Df(z)(h_{1}+h_{2})=Df(z)(h_{1})+Df(z)(h_{2})}$. As we also have ${\displaystyle Df(z)(th)=tDf(z)(h)}$, for t real, this mapping is linear. It is called the derivative of f(z) in the point z.

The linearity means that ${\displaystyle Df(z)}$ is determined by the two real numbers ${\displaystyle Df(z)(1)}$ and ${\displaystyle Df(z)(i)}$. These are denoted by ∂f(z)/∂x and ∂f(z)/∂y, respectively, and are called the partial derivatives with respect x and y. We have for ${\displaystyle h=h_{x}+ih_{y}}$:

Df(z)(hx + ihy) = (∂f(z)/∂x)hx + (∂f(z)/∂y)hy

This number is the scalar product of the vectors (∂f(z)/∂x, ∂f(z)/∂y) and (${\displaystyle h_{x},h_{y}}$), so, if we regard Df(z) and h as vectors, we can write:

Df(z)(h) = Df(z)*h.

The vector Df(z) is called the gradient of f(z) in the point z: the direction of Df(z) is the direction of the most rapid growth and the length of Df(z) is the growth of f(z) in this direction.

If ∂f(z)/∂x and ∂f(z)/∂y exist in a neighbourhood of z* and are continuous in z*, then f(z) is differentiable in z*.

The derivative of a mapping into the plane

If f(z) is a mapping from a domain of the plane into the plane, we can write ${\displaystyle f(z)=f_{x}(z)+if_{y}(z)}$, where ${\displaystyle f_{x}(z)}$ and ${\displaystyle f_{y}(z)}$ are real functions. f(z) is called differentiable in the point z (as real function), if both ${\displaystyle f_{x}(z)}$ and ${\displaystyle f_{y}(z)}$ are differentiable in z. If that is so, we have a linear mapping Df(z) from the complex plane into itself given by ${\displaystyle Df(z)(h)=Df_{x}(z)(h)+iDf_{y}(z)(h)}$. This linear mapping is called the derivative of f(z) in the point z.

The linearity means that Df(z) is determined by the two complex numbers Df(z)(1) (= ∂fx/∂x + i∂fy/∂x) and Df(z)(i) (= ∂fx/∂y + i∂fy/∂y), and we have:

Df(hx + ihy) = ((∂fx/∂x)hx + (∂fx/∂y)hy) + i((∂fy/∂x)hx + (∂fy/∂y)hy)

In matrix notation, this means that Df(z) is the linear mapping from the plane into itself given by

${\displaystyle {\begin{bmatrix}\partial f_{x}/\partial {x}&\partial f_{x}/\partial {y}\\\partial f_{y}/\partial {x}&\partial f_{y}/\partial {y}\\\end{bmatrix}}\times {\begin{bmatrix}h_{x}\\h_{y}\\\end{bmatrix}}}$

That f(z) is differentiable as a complex function, means that this multiplication corresponds to multiplication by the complex number f'(z), and this is the case precisely when the Cauchy-Riemann equations

${\displaystyle {\frac {\partial }{\partial {x}}}f_{x}={\frac {\partial }{\partial {y}}}f_{y}}$ and ${\displaystyle {\frac {\partial }{\partial {y}}}f_{x}=-{\frac {\partial }{\partial {x}}}f_{y}}$

are satisfied - these two number are the real and the imaginary part of f'(z).

Matrix calculus

A matrix is a rectangular array of real numbers. We will only need matrices of side 1 or 2. That is, either a 2x2-matrix (quadratic matrix):

${\displaystyle {\begin{bmatrix}a&b\\c&d\\\end{bmatrix}}}$

or a 1x2-matrix (a row-matrix): {a, b}, or a 2x1-matrix (a column-matrix):

${\displaystyle {\begin{bmatrix}a\\b\\\end{bmatrix}}}$

or a 1x1-matrix: {a}, identified with the number a.

The transpose of a matrix, is the matrix formed by reflection in the diagonal. This operation is denoted by *, and it means that we can denote a column-matrix by {a, b}* - the transpose of the row-matrix {a, b}.

Two matrices A and B of the same type can be added by adding the number on the corresponding places. We multiply a matrix by a real number by multiplying each of its elements by this number.

Two matrices A og B, where the width of A is equal to the height of B can be multiplied: the result is a matrix AB whose height is the height of A and whose width is the width of B:

${\displaystyle {\begin{bmatrix}a&b\\c&d\\\end{bmatrix}}\times {\begin{bmatrix}\alpha &\beta \\\gamma &\delta \\\end{bmatrix}}={\begin{bmatrix}a\alpha +b\gamma &a\beta +b\delta \\c\alpha +d\gamma &c\beta +d\delta \\\end{bmatrix}}}$
${\displaystyle {\begin{bmatrix}a&b\\c&d\\\end{bmatrix}}\times {\begin{bmatrix}\alpha \\\beta \\\end{bmatrix}}={\begin{bmatrix}a\alpha +b\beta \\c\alpha +d\beta \\\end{bmatrix}}}$

The product {a, b}{${\displaystyle \alpha ,\beta }$}* is the number ${\displaystyle a\alpha +b\beta }$ (the scalar product of the vectors {a, b} and {${\displaystyle \alpha ,\beta }$}) and the product {a, b}*{${\displaystyle \alpha ,\beta }$} is the 2x2-matrix

${\displaystyle {\begin{bmatrix}a\alpha &a\beta \\b\alpha &b\beta \\\end{bmatrix}}}$

The determinant of the quadratic matrix A =

${\displaystyle {\begin{bmatrix}a&b\\c&d\\\end{bmatrix}}}$

is the real number det(A) = |A| = ad - bc. The unit-matrix I is

${\displaystyle {\begin{bmatrix}1&0\\0&1\\\end{bmatrix}}}$

The inverse matrix to the 2x2-matrix A, is the 2x2-matrix ${\displaystyle A^{-1}}$ satisfying ${\displaystyle AA^{-1}=A^{-1}A=I}$. It is given by:

${\displaystyle {\begin{bmatrix}d&-b\\-c&a\\\end{bmatrix}}}$

divided by |A| - it does therefore only exist for |A| <> 0.

The points of the plane can be identified with the column matrices {x, y}*. Therefore, a 2x2-matrix A determines a linear mapping of the plane into itself: {x, y}* → A{x, y}*. It is injective (and then also bijective) if |A| <> 0. If that is so, the number |A| is the area of the image of the unit-square (if |A| is negative, the mapping changes orientation). Likewise, the vectors of the plane can be identified with the row-matrices {a, b}. A row-matrix {a, b} determines a linear mapping of the plane into the real numbers: {x, y}* → {a, b}{x, y}* = ax + by (the scalar product of the vectors {a, b} and {x, y}). The complex number z = x + iy can be identified with the column-matrix {x, y}* and with the 2x2-matrix

${\displaystyle {\begin{bmatrix}x&-y\\y&x\\\end{bmatrix}}}$

The mapping of the plane into itself given by this matrix, is the multiplication by the complex number z.