Complex number exponential

A very useful way to simplify Eq. (10.65) involves the complex number e in which i = / 1 equals cos y + i sin y. Therefore cos y is given by the real part of e y. Since exponential numbers are easy to manipulate, we can gain useful insight into the nature of the cosine term in Eq. (10.65) by working with this identity. Remembering that only the real part of the expression concerns us, we can write Eq. (10.65) as... 

Sufficient Conditions for the Existence of Laplace Transform Suppose/ is a function which is (1) piecewise continuous on eveiy finite intei val 0 < t exponential growth at infinity, and (3) Jo l/t)l dt exist (finite) for every finite 6 > 0. Then the Laplace transform of/exists for all complex numbers. s with sufficiently large real part. 

In order to describe the material properties as a function of frequency for a body that behaves as a Maxwell model we need to use the constitutive equation. This is given in Equation (4.8), which describes the relationship between the stress and the strain. It is most convenient to express the applied sinusoidal wave in the exponential form of complex number notation ... 

Recognizing that many chemistry students do not have a strong background in physics, I have introduced most of the chapters with some essential physics, concerning waves, mechanics, and electrostatics. I have also tried to keep the mathematical level at a minimum, consistent with a proper understanding of what is necessary. Basic calculus and an understanding of the properties of elementary trigonometical and exponential functions are assumed but I have not used complex numbers. Each chapter ends with some simple problems. 

Next, we can express the complex number in square brackets as an exponential, using this equality from complex number theory ... 

It is often convenient to use a complex number representation for sound waves [2]. The harmonic wave is represented by the complex exponential. 

Remember 1.4 The exponential representation of a complex number plays an important role in impedance analysis. Remember that — cos(0) j sin(0). 

Equation (2.13) is not yet in a form that is fundamentally different from the Cartesian form expressed in equation (2.5). However, we can obtain an alternative, more compact, and far more powerful way of writing the polar form of a complex number by re-visiting the Maclaurin series for the sind, cos6 and exponential functions. The Maclaurin series for cosine and sine are ... 

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% ... 

There is a theorem, known as Euler s formula, that allows a complex number to be written as an exponential with an imaginary exponent. 

The exponential function gives us a convenient way to express a complex number z in its polar form that is. 

There is a near-mystical expression of equality in mathematics known as Euler s Identity, which links trigonometry, exponential functions, and complex numbers in a single equation ... 

Converting the cosine/sine form to the complex exponential form allows many manipulations that would be very difficult otherwise (for an example, see Section 2 of Appendix A Convolution and DFT Properties). But, if you re totally uncomfortable with complex numbers and Euler s Identity (or with the identities of IS"" century mathematicians in general), then you can write the DFT in real number terms as a form of the Fourier Series ... 

The interpretation is the same as for the Fourier transform defined in section 10.1.6 For each required frequency value, we multiply the signal by a complex exponential waveform of that frequency, and sum the result over the time period. The result is a complex number which describes the magnitude and phase at that frequency. The process is repeated until the magnitude and phase is found for every fi-equency. 

The basic approach consists of defining on a purely mathematical basis a CPE by an impedance or an admittance expressed as a complex number proportional to a power p of the prodnct of the imaginary nnmber times the angular frequency (O, which is eqnivalent to say that the argnment of an exponential fnnction is a multiple of the same product... 

Insertion of this complex number into the argument of the exponential function featuring the wave function provides the expression of this latter ... 

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