Complex number argument

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. 

And the argument of the quotient of two complex numbers is the difTerence between the arguments. 

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. 

Now, the argument (phase angle) of the product of a series of complex numbers is the sum of the arguments of the individital numbers. 

We use common (but not universal) mathematical notation and terminology for functions. When we define a function, we indicate its domain (the objects it can accept as arguments), the target space (the kind of objects it puts out as values) and a rule for calculating the value from the argument. For example, if we wish to introduce a function f that takes a complex number to its absolute value squared, we write... 

We will show that Uhy = 0 by showing that for any a we have Tq,([//zy) = 0. By the argument above wc know that there is a complex number c such that Tahv = cTayhv and hence... 

Assuming that in the density matrix, p0r H0, and N (a number of particle operator) commute, then p0 can be rewritten in terms of the evolution operator U with the complex time argument ... 

If the real and imaginary parts of the complex quantity (number) Z are a function of a certain scalar argument, such as the angular velocity to, as co varies, the complex number Z traces a curve which is called... 

Determine the modulus and argument of a complex number, and denote its location on an Argand diagram... 

Figure 2.3 An Argand diagrann showing the complex numbers Zi= - 1 + i and Z2=1 - i with modulus V2 and arguments 3n/4 and -n/A, respectively...

Hint you will need to exercise a little care in determining the argument for the second of the two complex numbers. 

It may seem odd to think of the exponential function, z = e , as periodic because it is clearly not so when the exponent is real. However, the presence of the imaginary number i in the exponent allows us to define a modulus and argument as 1 and 6, respectively. If we represent the values of the function on an Argand diagram, we see that they lie on a circle of radius, r= 1, in the complex plane (see Figure 2.4). Different values of 6 then define the location of complex numbers of modulus unity on the circumference of the circle. We can also see that the function is periodic, with period 2% ... 

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