Relative Min of a Complex Number

[corbell777]

In calculus class, I learned that in order to find the relative minimum of a fxn, I would take the first derivative of the fxn, then look for where the graph went from negative to positive. The point at which it did so would be a relative minimum. But now I want to find the relative minimum of a function made up of complex numbers. In order to find the relative minimum, would I now be looking for a saddle point? How could I tell the difference between a relative minimum, a relative maximum, and a point of inflection?

 

[HallsofIvy]

In calculus class, I learned that in order to find the relative minimum of a fxn, I would take the first derivative of the fxn, then look for where the graph went from negative to positive. The point at which it did so would be a relative minimum. But now I want to find the relative minimum of a function made up of complex numbers. In order to find the relative minimum, would I now be looking for a saddle point? How could I tell the difference between a relative minimum, a relative maximum, and a point of inflection?

First, the set of complex numbers is NOT an "ordered" set so I presume you are talking about a real valued function of a complex variable. You can write the the function of a complex variable z= x+ iy as a function of two real variables, f(z)= f(x, y). You use the term "saddle point" so apparently you know a little about functions of two variables. But, no, a "saddle point" is not a minimum. You can find critical points, which might be minima, maxima, or saddle points by finding points where the two partial derivatives, \(\displaystyle f_x\) and \(\displaystyle f_y\), are 0. You can determine which by using an extension of the "second derivative test". Look at \(\displaystyle f_{xx}f_{yy}- f_{xy}^2\) (this is the determinant of the matrix \(\displaystyle \begin{bmatrix}f_{xx} & f_{xy} \\ f_{xy} & f_{yy}\end{bmatrix}\)). If that is negative, you have a saddle point (so neither a minimum not a maximum). If it is positive, you have either a maximum or a minimum and can tell by looking at \(\displaystyle f_{xx}\) or \(\displaystyle f_{yy}\). They must both have the same sign. If they are positive, you have a minimum, if they are negative, a maximum.