The number of zeros will grow and can accumulate in areas, curves and points of the

complex plane. Especially in the thermodynamic limit a pair of zeros can become closer to the positive real axis.

The fourth edition adds video solutions for selected exercises on the companion website, a section on the

complex plane, and examples of finding the magnitude of a scalar multiple, multiplying in the

complex plane, and using matrices to transform vectors.

Wavenumbers at frequency of 1 MHz (represented by dashed line) on the

complex plane come out as symmetric with respect to the axes of the

complex plane.

The tasks of the kinematic research of

complex plane mechanisms remain relevant because in each particular case of the upper class mechanism research it is necessary to select and execute an original sequence of actions, which is caused by the simultaneous application of several methods of kinematic analysis, while a general method for studying the variety of such complex mechanisms of the fourth and higher classes does not exist at the present moment.

The expansions (1.3), (1.4), and (1.5) have the good property of being given in terms of elementary functions of z, but they have the inconvenience of not being uniform in |z| in unbounded regions of the

complex plane that include the point z = 0.

For the three elementary surfaces, we will construct the map j : [summation] right arrow] [??] = [S.sup.2] by first diffeomorphically (but not biholomorphically!) identifying [summation] with the

complex plane C of complex coordinate [zeta] (or with C with a point removed) and then identifying the latter with the once- or twice-punctured sphere by using stereographic projection from the north pole of [S.sup.2]:

Henceforth, unless specified otherwise, A is supposed to be a scalar type spectral operator in a complex Banach space (X, [parallel]*[parallel]) with strongly o-additive spectral measure (the resolution of the identity) [E.sub.A](*) assigning to each Borel set [delta] of the

complex plane C a projection operator [E.sub.A]([delta]) on X and having the operator's spectrum [sigma](A) as its support [7, 8].

It is clear that [F.sup.2.sub.1] (C) is the classical Fock space on the

complex plane C, which is also denoted by [F.sup.2](C).

Since L-function itself can be analytically continued as a meromorphic function in the whole

complex plane, therefore, L-functions will be taken as special meromorphic functions, with the help of Nevanlinna's value distribution theory, in order to study the uniqueness of L-functions.

The plane on which imaginary (and complex) numbers are plotted is called an Argand Diagram or the

Complex Plane. On it, the horizontal axis is called the real axis and the perpendicular vertical axis is called the imaginary axis.

Value distribution of L-functions concerns distribution of zeros of L-functions L and, more generally, the c-points of L, i.e., the roots of the equation L(s) = c, or the points in the pre-image [L.sup.-1] = {s [member of] C : L(s) = c}, where and in what follows, s denotes a complex variable in the

complex plane C and c denotes a value in the extended

complex plane C [union] {[infinity]}.