Graphical interpretation and visualizing the Complex Roots of Transcendental functions with Real Coefficients.
Enter the values of the coefficients a and b of the Transcendental function and find its Roots.
Graphical interpretation the Roots: the intersection of implicit functions, which are the zeroed real and imaginary parts of the complex function f(z), respectively: real(f(z))=0 and imaginary(f(z))=0.
In the general case, these roots can be found numerically. Here, you can set the function f(x). Using 9 test points zi, approximately find its roots by moving them to the intersection points of the implicit functions under consideration, and make sure that in these cases fzi=f(zi) is at the origin, i.e. zi are the roots of the original equation. The roots and function values for them are shown in the table.
The z8 and z9 complex numbers are at the intersection of the considered implicit functions. However, they are not the roots of the equation: f(z8)≠0 and f(z9)≠0!? Either the graphical representation of the implicit functions is not accurate, or... The gradients of both functions at these points are very large and a great precision of calculating the intersection points is required: Points on the "real" and "imaginary" curves, respectively Point(eqa) and Point(eqb) near z8(or z9), nullify these equations!
Another example can be found in the applet https://www.geogebra.org/material/show/id/gamqfzzw
*New version available.