Polar representation of a complex number

Making use of the polar representation of a complex number, the nuclear wave function can be written as a product of a real amplitude, A, and a real phase, S,... 

While the FT of the even cosine function is real, the result for sin(27rvot) is imaginary (S(v — vo)- --b Vo))/(2i). Since the sine function is 90° out of phase to the corresponding cosine, it is clear that the imaginary axis is used to keep track of phase shifts, consistent with the polar representation of a complex number x + iy = re with r = Jx - -y and 4> = tan (y/x). In this phasor picture, positive and negative frequencies can be interpreted as clockwise and counterclockwise rotations in the complex plane. 

The representation of an Ohmic impedance as a complex number represents a departure from standard practice. As will be shown in subsequent sections, the local impedance has inductive features that are not seen in the local interfacial impedance. As the calculations assumed an ideally polarized blocking electrode, the result is not influenced by Faradaic reactions and can be attributed only to the Ohmic contribution of the electrolyte. 

The evaluation of log A is thus carried out in terms of the complex scalar logarithmic function. For a complex number z, the logarithm is obtained from the polar representation... 

This is one of the most important formulas in physics. One of the reasons is that it gives support to the graphical representation of complex numbers in polar coordinates, as illustrated in Figure 9.9 with the wave function represented by a vector having the semienergies as components. 

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