TABLE OF CONTENTS The purpose of this appendix is to give a review of the theory of functions of a complex variable that we shall need in the discussion of conformal mapping of a microstrip transmission line onto an ideal parallel plate guiding structure, which has less complicated boundary conditions, and therefore makes transmission line analysis much simpler.
The theory of functions with a complex variable is well described in numerous publications [1 6]. In the theory of complex variables, we use the complex variable
| (A.1) |  |
where both x and y are real variables. The values of the independent variables x and y cover the total plane spanned by the coordinates. Any given value of z is associated with a point in the x- y-plane, usually termed complex the z -plane. The complex number z also may be expressed by polar coordinates in terms of r and ?:
| (A.2) |  |
with the magnitude of z, denoted by z. ? is called the argument of z, and denoted by arg z. Hence, we can write z in polar form
| (A.3) |  |
Note, however, that for a given z, arg z is not unique. Clearly, we can add any integer multiple of 2 ? to ? without affecting the value of the function z = x + jy. If ? satisfies the equation - ?
