Pictures of Julia and Mandelbrot Sets/Terminology

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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 f(x, y) = f_x(x, y) + if_y(x, y) of the plane into itself: the two equations

\frac{\partial}{\partial{x}}f_x = \frac{\partial}{\partial{y}}f_y and \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 i^2 = -1

convergence: a sequence 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 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 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

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 {a_{ij}}: the real number |{a_{ij}}| = 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): z_1 = f(z_0), z_2 = f(z_1), z_3 = f(z_2), ...

lim: if the sequence 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 {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 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

\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

\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 a_0 + a_1z + a_2z^{2} + ... + a_nz^{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 v_1 = {x_1, y_1} and v_2 = {x_2, y_2}: the real number v_1*v_2 = x_1x_2 + y_1y_2 = |v_1||v_2|cos(\theta), where \theta is the angle between v_1 and v_2

transcendental function: a function that cannot be constructed in a finite number of steps from elementary functions and their inverses, e.g. 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 z = x+iy is 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 |z|^2 = zz^-. From this, we can derive the rule for division: z/w = zw^-/(ww^-) = zw^-/|w|^2.

A complex number z = x + iy can be written

z = r(sin(\theta) + icos(\theta))

where r is the norm of z:

r = |z| = √(x2 + y2)

and where the angle \theta is the argument arg(z) of z:

\theta = arctan(y/x) for x > 0
\theta = arctan(y/x) + \pi for x < 0

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

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

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

The sinus and cosinus relations can be written

cos(\theta_1+\theta_2) + i sin(\theta_1+\theta_2) = (cos\theta_1 + i sin\theta_1)(cos\theta_2 + i sin\theta_2)

and from this we see that e^z has the exponential property for z imaginary:

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: log(z) = w + n2\pi i, for every integer n.

For a positive real number a and a complex number z, we define a^z = e^{log(a)z}. For a complex number z, the power 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:

d(f(z) + g(z))/dz = df(z)/dz + dg(z)/dz

d(f(z)g(z))/dz = (df(z)/dz)g(z) + f(z)(dg(z)/dz)

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

d z^n/dz = nz^{n-1} (n integer)

d a^z/dz = log(a)a^z (a positive real number)

d(exp(z))/dz = exp(z)

d(log(z))/dz = 1/z

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

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

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 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 Df(z)(h) from the complex numbers (h) into the real numbers satisfies Df(z)(h_1 + h_2) = Df(z)(h_1) + Df(z)(h_2). As we also have Df(z)(th) = t Df(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 Df(z) is determined by the two real numbers Df(z)(1) and 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 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 (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 f(z) = f_x(z) + if_y(z), where f_x(z) and f_y(z) are real functions. f(z) is called differentiable in the point z (as real function), if both f_x(z) and 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 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

\begin{bmatrix}
\part f_x/\part{x} & \part f_x/\part{y} \\
\part f_y/\part{x} & \part f_y/\part{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

\frac{\partial}{\partial{x}}f_x = \frac{\partial}{\partial{y}}f_y and \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):


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

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


\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:

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


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

The determinant of the quadratic matrix A =


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

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


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

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


\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


\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.