Complex number magnitude

The two factors on the right are both positive, real numbers less than one. If the magnitudes of U(h and U h ) are both close to one, therefore, the magnitude of the difference between the temis within the brackets on the left (complex numbers in general) must be small. 

To analyse the response of this circuit to an alternating voltage, it turns out to be rather easier to replace the simple sinusoidal form of the voltage used above by V = V e ", where the complex number e,w = cos wt + t sin wt and i = J — 1 Any component of a circuit such as that shown can be defined as having an impedance Z, which can also be thought of as a complex number, containing both phase and magnitude information. For a resistor, Z is entirely real and simply equal to the resistance R, but for a capacitor ... 

When expressed in polar coordinates, the quantity r is the magnitude or absolute value or modulus and (f> is the argument or phase of the complex number. It follows immediately that... 

This is a complex number with magnitude of one and argument equal to —ciiD. 

This example illustrates a very important property of complex numbers. The magnitude of the product of two complex numbers is the product of the magnitudes of each. The argument of the product of two complex numbers is the sum of arguments of each. 

The best model and the power n is determined by selecting the form that best satisfies the equations given below. These equations come from two equations that give the argument and the magnitude of the complex number. 

Since spherical harmonics are functions from the sphere to the complex numbers, it is not immediately obvious how to visualize them. One method is to draw the domain, marking the sphere with information about the value of the function at various points. See Figure 1.8. Another way to visualize spherical harmonics is to draw polar graphs of the Legendre functions. See Figure 1.9. Note that for any , m we have F ,m = , the Legendre function carries all the information about the magnitude of the spherical harmonic. 

F. Consequently, the scattering amplitude, e, is a complex number whose amplitude reflects the scattering strength, and whose phase factor determines the constructive and destructive interference conditions for any particular value of Q. The diffraction intensities are proportional to the magnitude of the scattering amplitudes. Consequently the intensities do not depend upon an absolute phase, but instead on the relative phases for each scattering event. 

The ratio > o/ o the phase shift can be obtained from the transfer function by inserting s=jo) and evaluating the magnitude and phase of the resulting complex number G(ja)) ... 

Any complex number, F, can be represented in a complex plane diagram, as illustrated (Figure 6.5), and so takes on the appearance of a vector with the twin properties of magnitude and direction. Magnitude is defined by the length, F, and direction by the angle, a, subtended... 

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