# 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 of the plane into itself: the two equations

- and ,

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

**convergence**: a sequence (*i* = 0, 1, 2, ...) converges to the point *z**, if for each neighbourhood *U* of *z**, there exists a number N such that 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 (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

(h complex) if it exists

**determinant** of the 2x2 matrix {}: the real number |{}| =

**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):

**lim**: if the sequence (*n* = 0, 1, 2, ...) converges to the number *a*, we write *lim _{n → ∞} a_{n} = a*, if the value of the function

*f(z)*converges to

*a*for

*z*converging to

*z**, we write

*lim*

_{n → ∞}f(z) = a**matrix**: a rectangular array of numbers {} (*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 , 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| = √(x ^{2} + y^{2})* ≥ 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

where h is real (if it exists)

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

where h is real (if it exists)

**polynomial** of degree n: a function of the form

**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 = {, } and = {, }: the real number , where is the angle between and

**transcendental function**: a function that cannot be constructed in a finite number of steps from elementary functions and their inverses, e.g. , 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**

*i*^{2}= -1*(x*_{1}+ iy_{1}) + (x_{2}+ iy_{2}) = (x_{1}+ x_{2}) + i(y_{1}+ y_{2})*(x*_{1}+ iy_{1})(x_{2}+ iy_{2}) = (x_{1}x_{2}- y_{1}y_{2}) + i(x_{1}y_{2}+ x_{2}y_{1})*(x*_{1}+ iy_{1})/(x_{2}+ iy_{2}) = ((x_{1}x_{2}+ y_{1}y_{2}) + i(-x_{1}y_{2}+ x_{2}y_{1}))/ (x_{2}^{2}+ y_{2}^{2})

The **conjugate** number to is , that is, the reflection of *z* in the x-axis. The **norm** of *z* is the real number *|z| = √(x ^{2} + y^{2})*. We have . From this, we can derive the rule for division: .

A complex number *z = x + iy* can be written

where *r* is the norm of *z*:

*r = |z| = √(x*^{2}+ y^{2})

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

- =
*arctan(y/x) for x > 0* - =
*arctan(y/x) + for x < 0*

*arg(z)* is multivalued: *arg(z)* = , for every integer *n*.

The point (lying on the unit circle) is also denoted , and we define the **exponential function** by

The *sinus and cosinus relations* can be written

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

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: , for every integer *n*.

For a positive real number *a* and a complex number *z*, we define . For a complex number *z*, the power 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:

(*n* integer)

(*a* positive real number)

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

For the computer it is easy to find *f'(z)*: *f'(z) = (f(z+h) - f(z))/h* for , 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

*lim*_{t → 0}(f(z + th) - f(z))/t

(t real) exists for every complex number h, and if the hereby defined function from the complex numbers (h) into the real numbers satisfies . As we also have , for t real, this mapping is *linear*. It is called the derivative of f(z) in the point z.

The linearity means that is determined by the two real numbers and . 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 :

*Df(z)(h*_{x}+ ih_{y}) = (∂f(z)/∂x)h_{x}+ (∂f(z)/∂y)h_{y}

This number is the scalar product of the vectors *(∂f(z)/∂x, ∂f(z)/∂y)* and (), 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 , where and are real functions. *f(z)* is called differentiable in the point *z* (as real function), if both and are differentiable in *z*. If that is so, we have a linear mapping *Df(z)* from the complex plane into itself given by . 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) (= ∂f _{x}/∂x + i∂f_{y}/∂x)* and

*Df(z)(i) (= ∂f*, and we have:

_{x}/∂y + i∂f_{y}/∂y)*Df(h*_{x}+ ih_{y}) = ((∂f_{x}/∂x)h_{x}+ (∂f_{x}/∂y)h_{y}) + i((∂f_{y}/∂x)h_{x}+ (∂f_{y}/∂y)h_{y})

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

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

- and

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):

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

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*:

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

The **determinant** of the quadratic matrix *A* =

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

The **inverse matrix** to the 2x2-matrix *A*, is the 2x2-matrix satisfying . It is given by:

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

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