Where do imaginary solutions hide in the graph?

Forum|Forum|2 years ago
June 12, 2023
2 replies
1944 views

Hello,

The question is related to finding of solutions of the below equation:

Solution with solve:

Solution with root:

From the graph we see that the graph cuts the x-axis only in 2 points. Where can we see the other 2 solutions (which are the imaginary solutions) with the root solution on the graph that solve gives above? Or can't we seen on the graph these 2 imaginary solutions of f(x) with the root function/solution?
How to find with root function the imaginary solution of the function f(x)?

Mathcad Prime 8 file attached.
Thank you.

1_Imaginary_Solution_On_Graph.zip

Calculus_Derivatives

Best answer by Werner_E

BTW, a non-real number is not necessary imaginary (as this would mean its real part is zero).

In the 2D plot you can only "see" points with real valued coordinates.

To see the non-real ones you would need a 4D-plot, where the x-"axis" as well as the y-"axis" are Gauß-planes.

Could be a nice feature suggestion for prime 😉

As a workaround you could use two 3D-plots with x=real part of argument, y=imaginary part of argument and z=real part of function value or imag. part of function value.

W

Werner_E

Answer

25-Diamond I

BTW, a non-real number is not necessary imaginary (as this would mean its real part is zero).

In the 2D plot you can only "see" points with real valued coordinates.

To see the non-real ones you would need a 4D-plot, where the x-"axis" as well as the y-"axis" are Gauß-planes.

Could be a nice feature suggestion for prime 😉

As a workaround you could use two 3D-plots with x=real part of argument, y=imaginary part of argument and z=real part of function value or imag. part of function value.

W

Werner_E

25-Diamond I

Here in more detail, using implicit2d() from Viacheslav N. Mezentsev

The green curve is the intersection of the green surface (real parts of function values) with the a-b-plane (function is evaluated for a+b*i).

The orange curve is the intersection of the orange surface (imaginary parts of the function values) with the a-b-plane.

The four solutions of your equation can be found in the intersection of both curves.

The a-b-plane is of course the Gauß-plane of the complex function arguments.

Of course the four solutions also are part of the intersection of the green and orange surface (points where real and imaginary part of function value are equal, black line), but I found the plot too much overloaded to be informative. Maybe, if your would have some degree of transparency, but Prime does not have any means to make a surface semi-transparent.