Open source cross-platform software of geometry, analysis and simulation - Yves Biton
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Calculus and representation of complex numbers with MathGraph32

publication Monday 31 January 2011.

MathGraph32 allows to calculate and represent complexe numbers.

You can use usual transcendant function on complex numbers, create a point via it’s affix in a frame, mesure the complex affix of a point in a frame, create complexe functions of one, two or three complex variables, recurrent sequences kind of $u_{n+1} = f(u_n)$ and graph these sequences.

A complex calculus can use every calculus or measure (real or complex) defined before.

If you want to use in a real calculus the real part, imaginary part, argument or module of a complex number, you must first create a real calculus via menu items calculus - New Real calculus - Real part of a complex or calculus - New Real calculus - Imaginary part of a complex or calculus - New Real calculus - Principal complex argument or calculus - New Real calculus - Complex modulus.

Example : Vous wish to visualize the image of the circle center O and radius 1 through the complex transformation associated to function defined by f(z) = z + 1/z^n where n est an integer lying between 1 and 10.

Use menu item File _ New Figure with - Frame with vectors and choose an orthonormal frame and vectors named u and v.

Use icon to create the circle of center O and through point I(point I is the edge of vector u. It’s name is hidden).

Use menu item Calculus - New variable to create a variable named n with 1 as mini value, 10 as maxi value, 1 as step and 2 as current value. Set the Associated dialog checkbox selected.

Use menu item Calculus - New complex calculus - Complex function to create a complex function of one varible named t with formula t+1/t^n.

Use menu item Macro - New macro - Variable increment. Click on the top left corner of the figure.

A dialog box pops up. In the field Title enter Increment n. Variable n is already selected. Validate with OK.

In the same way, create a macro decrementing variable n under the former one with menu item Macro - New macro - Variable decrement.

Use icon to create a value display of variable n (vous pouvez utiliser le bouton Liste des valeurs pour sélectionner n). In the Starting string editor enter n=.

With icon create a linked point to the circle previously created and use l’icone to give name M to this point.

Use menu Calculus - Measure - Point affix or icon to measure affix of point M (just click on M). This measure is represented as Aff(M,O,I,J).

Via menu item Calculus - New complex calculus >> Complex calculus, create a calculus named z with formula Aff(M,O,I,J) (You can use button Values ).

In the same way create a complex calculus named z’ with formula f(z).

Use icon to create point with affix z’ and name this point M’.

Now we have to create the locus iof point M’ generated by the positions of point M on the circle.

In the color palette, activate the red color.

Now use icon . Click first on point M’ then on M. In the dialog box popping up, set the checkbox Closed locus selected et ask for 300 points. Validate.

The figure is now ready.

You can see this figure underneath, activated by MathGraph32 JavaScript library.