JPlotter

The Options menu provides three different settings for the display of complex values: No Complex Values: this is the default case for non-complex functions. However if the function contains complex numbers, they will be treated as NaN (not a number) and thus not be plotted. Real and Imaginary Part: this is the default case for functions that contain at least one complex number. A complex number is `y=a+b*i` (with `i` as the imaginary unit) where `b!=0`. Amplitude and Phase: this is another possible view for complex numbers: `y=|y|*e^(i*arg(y))=sqrt(a^2+b^2)*e^(i*arctan(a/b))`.

Real and Imaginary Part:

Amplitude and Phase:

No Complex Values:

With complex numbers disabled the part of the function with complex numbers cannot be plotted. So for `x<0` the values are NaN (not a number): `sqrt(x)={(sqrt(x),if x>=0),(text{undefined},if x<0):}`

Example: RCR circuit

The imaginary unit `i` can be directly used in the function Let our complex function be `(1+i*x)/(1+2*i*x)` This is the system equation of a RCR system: (where `R=2` and `C=1/2`).

The function wizard allows for adding special functions and provides templates for easier creation of mathematical functions f(x). The template Parabola was initially a test for the template functionality. The parabola equation can directly be typed into the function text field without the use of this wizard. The Any Function template provides a list of all mathematical functions supported from which any custom function f(x) can be built. However once the syntax and the available functions are known the desired function f(x) can also directly be typed into the function text field without the use of this wizard.

Rectangle, Sawtooth, Unity step function:

Rectangle function with the following properties: `y_max = 1` `y_min = 0` `Delta x_1 = 1` `Delta x_2 = 1` non-periodic (single pulse) Interval: `[-1 ; 2]` The real DFT part is a high-frequent si-function (`si(x)=sin(x)/x` starting from left to middle, then another interval starts). The imaginary DTF part is basically zero. Rounding errors cause a noisy function. However the y-axis scale displays all zeros.

High-frequent si function `si(x)+si(x-200)` within `[0;200]`. The DFT is a rectangle again. Since `F{F{f(x)}} = f(-x)` the rectangle is inverted. The imaginary part again is a noise function caused by rounding errors.

The plot shows: Sawtooth function with: `y_max = 1` ; `y_min = 0` ; `Delta x_1 = 2` ; `Delta x_2 = 0` Unity step function with: `y_0 = 0` ; `A = 1` ; `Delta x = 0` Unity step function with: `y_0 = 0.1` ; `A = 0.9` ; `Delta x = -3` Unity step function with: `y_0 = 0.8` ; `A = 0.5` ; `Delta x = 3`

Dirac Impulse:

The Dirac Impulse is a very short pulse of theoretically infinite height. The Dirac Impulse contains all frequencies over the complete domain. The imaginary part again is a noise function caused by rounding errors.

Interpolation of points:

Example: Interpolation of 8 points: `0;1` `1;-1.5` `2;2` `3;1.5` `4;3` `5;0` `6;0` `7;1.5`

Three interpolation techniques are displayed: Holding the value till the next value Linear interpolation Quadratic spline interpolation Other techniques are: Holding the value half way to the next/previous value. So each point represents the middle of a line. Quadratic interpolation between three points. This produces a set of separate parabolas without a continuous intersection at the edges.

The options are: Convolution: convolve the the selected function with itself or another function Hilbert transformation: perform Hilbert transformation on the selected function Derive function: plot the derivative of the selected function Integrate function: plot the integrated of the selected function Mirror function: mirror the selected function at x/y-axis Shift function: shift the selected function along the y-axis Calculate the area between the currently selected function and another function or the x-axis Find extrema / zeros of the currently selected function Change the appearance of the currently selected function (hightlight, hide or color change) or remove the function

Convolution:

Rectangle function with the following properties: `y_max = 1` `y_min = 0` `Delta x_1 = 1` `Delta x_2 = 1` non-periodic (single pulse) Interval: `[-0.5 ; 1.5]` Convolution with itself: The resulting function is a triangle function (real part). The imaginary part is noise, caused by rounding errors in the DFT.

Hilbert Transformation:

Rectangle function with the following properties: `y_max = 1` `y_min = 0` `Delta x_1 = 1` `Delta x_2 = 1` periodic Interval: `[-3 ; 3]` Hilbert Transformation in red. Due to rounding errors caused by the DFT the resultung function contains some noise. The imaginary part is theoretically constant.

Derive Function:

	`y_1 = x^3 + 2 x^2 + x - 8` `y_2 = y_1' = 3 x^2 + 4x + 1` `y_3 = y_2' = y_1'' = 6x + 4` `y_4 = y_3' = y_1''' = 6`		`y_1 = sin(x)` `y_2 = y_1' = cos(x)`
	`y_1 = si(x) = sin(x)/x` `y_2 = y_1' = (cos(x) * x - sin(x)) / x^2` `y_3 = y_2' = y_1'' = (-x^2 * sin(x) - 2x * cos(x) + 2 sin(x)) / x^3`

Integrate Function:

`y_1 = x^2` within `[-2 ; 2]` Mathematically: `int x^2 dx = 1/3 x^3 + C` `y_2 = int y_1 dx` with `f(x_min) = f(-2) = 0` to determine `C`: `f(-2) = 0 = 1/3 (-2)^3 + C => C = 8/3`. `y_3 = int y_1 dx` with `f(x_min) = f(-2) = -8/3` to determine `C`: `f(-2) = -8/3 = 1/3 (-2)^3 + C => C = 0`.

Mirror Function:

Mirror at x-axis: Any function plotted can be mirrored at the x-axis. The resulting function is `-f(x)`. Of cause manually added functions can be mirrored manually by just adding the same function with prefix '-'. However resulting functions (such as derivatives, Fourier transform, Hilbert transform, ...) can be mirrored by use of this option.		Example: `y_1 = x^2` `y_2 = H{y_1}` (Hilbert transformation) `y_3 = -y_2 = -H{y_1}`
Mirror at y-axis: Original user functions can be mirrored at the y-axis. The resulting function is `f(-x)`. However this option is not available for resulting functions (such as derivatives, Hilbert transform, ...). This is because resulting functions are calculated based on another function and mirroring would require data which is not there. For user functions `x` can be replaced by `-x` to aquire mirroring.		Example: `y_1 = sin(x)` `y_2 = sin(-x)`

Shift Function:

`y_1 = x^2` `y_2 = H{y_1}` (Hilbert transformation) `y_3 = y_2 + Delta y` (with `Delta y = 2`)

Combine Function:

	`y_1`: rectangle: `y_max = 1; y_min = 0; Delta x_1 = pi; Delta x_2 = pi;` non-periodic (single pulse); function is hidden `y_2 = sin(x)`; function is hidden `y_3 = y_1 * y_2`     the result is one single sine pulse.
	`y_1`: delayed unity step: `y_0 = 0; A = 1; Delta x = -1`; function is hidden `y_2 = x^2`; function is hidden `y_3 = y_1 * y_2`     the result is a parabola which starts at -1.

Area Calculation:

	Area between `x^2-2x` and the x-axis within `[-2;3]`: `~~ 9.333` Let's check: `A = int_(-2)^0 (x^2-2x)dx + |int_0^2 (x^2-2x)dx| + int_2^3 (x^2-2x)dx` `A = [1/3 x^3-x^2]_(-2)^0 + |[1/3 x^3-x^2]_(0)^2| + [1/3 x^3-x^2]_(2)^3` `A = -(1/3 (-2)^3-(-2)^2) + |1/3 2^3-2^2| + (1/3 3^3-3^2-(1/3 2^3-2^2))` `A = -(-8/3-4) + |8/3-4| + (27/3-9-(8/3-4))` `A = 20/3 + 4/3 + 4/3 = 28/3 ~~ 9.333`
	Area between `x^2-2x` and `sin(x)`: `~~ 11.89`

Find Extrema / Zeros:

	Extrema: Minima: x=0.467 ; y=-0.626 x=1.100 ; y=-0.333 x=1.733 ; y=-0.176 ... Maxima: x=0.133 ; y=0.851 x=0.767 ; y=0.456 x=1.400 ; y=0.244 ...
	Zeros (function crosses x-axis): x=0.314 x=0.628 x=0.943 x=1.257 x=1.571 ... To calculate the zeros of a polynomial (also complex zeros) see the Calculate Zeros tool.

The difference between a plot and a function is, that a plot is always a two-dimensional graph. So a function can have one or two plots (e.g. real and imaginary part). The plot options are not among the function options but below the list of value pairs in the Details window.

Apply a tangent on a function:

Check box Apply tangent on the desired plot tab. Moving the cursor over the canvas will apply a tangent at this very position. Click on the left mouse button will add the current tangent as a permanent function.

Example: Tangent on `sin(x)` at `x ~~ 2`.

Phase Portrait:

The result is obviously a circle. Lets understand why: The sine at `x=0` starts with a high positive slope but the y value is 0. Remember that y value is on the x-axis and slope is one the y-axis. This leads to point 1. Til `x=pi/2` the y value of sine increases while the slope decreases. At `x=pi/2` the y value is at its max value of 1, while the slope is 0. This leads to point 2. Then the y value of sine starts to fall as the slope turns negative. At `x=pi` the y value of sine is 0 while the slope has its max negative value (point 3). Then to `x=3pi/2` the slope decreases (while still negative) til 0 while the y value reaches its max negative value of -1 (point 4). The sine completes its period until `x=2pi` while the slope starts to increase again and the y value develops to 0. This is again at point 1.

The result is obviously a spiral, going from the edge to its center. Based on example 1 with the sine this is easy to understand. Basically we have a circle, too. But since the y value (and the slope as well) of the original function decreases each period the circle is a spiral becoming smaller and smaller.

The Tools menu provides different tools for mathematical calculations: Solve linear equation systems Calculate roots (zeros) of a polynomial Analyze a vector Analyze a matrix Pocket calculator: Rechne Console

Solve Linear Equation Systems:

	`((3,4),(8,-2)) * vec x = ((18),(10))` Solution: `vec x = ((2),(3))`
	`((1,2,1),(0,5,0),(0,3,-2)) * vec x = ((11),(20),(8))` Solution: `vec x = ((1),(4),(2))`
	`((2,6,-3,12),(4,3,3,15),(4,-3,6,6),(0,-3,5,-2)) * vec x = ((-6),(6),(6),(14))` Solution: `vec x = ((-3),(2),(4),(0))`

Calculate Zeros (Roots) of a Polynomial:

	`y = 3x^3 - 8x^2 - 11x + 10`	3 real zeros: `x_0 = 1+sqrt(6) ~~ 3.45` `x_1 = 2/3 ~~ 0.66` `x_2 = 1-sqrt(6) ~~ -1.45`
	`y = x^3 + x^2 + x + 1`	1 real, 2 complex zeros: `x_0 = -1` `x_1 = i` `x_2 = -i` Due to rounding errors the displayed results differ from the exact results.
	`y = x^2 + 4`	2 complex zeros: `x_0 = 2i` `x_1 = -2i` Proof: `0 = x^2 + 4 -> -4 = x^2` `x = +- sqrt(-4) = +- 2i`

Analyze a Vector:

• DFT: Discrete Fourier Transformation (see also section DFT). The result is a vector `F{vec x}` of the same lenght.
• Inverse DFT: inverse Discrete Fourier Transformation. The result is a vector `F^-1 {vec x}` of the same lenght.
• Different vector norms. The result for each norm is a real number.

	`F ((1),(4),(0),(-1)) = ((4),(1+5i),(-2),(1-5i))` `F^-1 ((1),(4),(0),(-1)) = ((1),(1/4-5/4i),(-1/2),(1/4+5/4i))`	Maximum norm: `max|x_i| = 4` Row sum norm: `sum_(i=1)^n |x_i| = 6` Euclidic norm: `sqrt(sum_(i=1)^n x_i ^2) = sqrt(18) ~~ 4.24`
The following vector represents 4 samples of the sine function. For comparison see section DFT for the graphical Fourier analysis of `sin(x)`. The result is one positive and one negative imaginary peak.
	`F ((0),(1),(0),(-1)) = ((0),(2i),(0),(-2i))` `F^-1 ((0),(1),(0),(-1)) = ((0),(-1/2i),(0),(1/2i))`	Maximum norm: `max|x_i| = 1` Row sum norm: `sum_(i=1)^n |x_i| = 2` Euclidic norm: `sqrt(sum_(i=1)^n x_i ^2) = sqrt(2) ~~ 1.41`

Analyze a Matrix:

• Calculate the determinant of a square matrix. The result is a real number.
• Calculate the Eigen values of square matrices up to size 3. The result is a vector of the Eigen values (number of Eigen values equals dimension of the matrix: 2 or 3).
• Invert a square matrix. The result is a new matrix of the same size.
• Different matrix norms. The result for each norm is a real number.

Eigen values: `det((3-lambda,-1,0),(2,-lambda,0),(-2,2-lambda,-1)) = 0 => {: (lambda_0=2),(lambda_1=1),(lambda_2=-1) :}` Determinant: `det A = -2`

`A = ((3,-1,0),(2,0,0),(-2,2,-1)) => A^-1 = ((0,1/2,0),(-1,3/2,0),(-2,2,-1))` Norms: Total norm: `n* max|a_(ij)| = 3*3 = 9` Row sum norm: `max_(1<=i<=n) sum_(j=1)^n |a_(ij)| = 5` Column sum norm: `max_(1<=j<=n) sum_(i=1)^n |a_(ij)| = 7` Frobenius norm: `sqrt( sum_(i=1)^n sum_(j=1)^n a_(ij)^2 ) = sqrt(23) ~~ 4.796`

Rechne Console:

Some Examples: 2*sin(1.5*Pi) # sine function and use of brackets lg(1.7)+4 # decimal logarithm sqrt(3+cos(2*Pi)) # square root, cosine and use of brackets 3rt(27) # 3rd (cubic) root; any other root also possible ANS^2 # reuse of the last result ln(e^(2^(2^2))) # natural logarithm and Eulers number (2*i-3)^2 # calculating with complex numbers factor(3425643) # factorization binom(17,9) # binominal coefficient gcd(2355,75,65) # greatest common divisor sum(x,1,10,x^2) # summation of x^2 with x from 1 to 10 ? # manual page with list of all supported functions/operations and examples

• Verbosity: if checked all intermediate calculation steps are displayed.
• Transfer between different numerical systems (e.g. decimal numbers to binary numbers; hexadecimal to decimal; ...)
• Precision: automatic (scientific display of numbers, use of E if appropriate), rounding to a fixed number of decimal places or rounding to integers.