Complex numbers expose all the roots of real (and complex) numbers.
For example, there should be two square-roots, three cube-roots, etc. of any number.Find the following roots.
What are the cube-roots of 27? In other words, what is
$27^{\left(\frac{1}{3}\right)}$ ?
What are the fifth roots of 3 (
$3^{\left(\frac{1}{5}\right)}$ )?
What are the fourth roots of one?
Cool exponentials
Simplify the following (cool) expressions.
$i^{i}$
$i^{(2i)}$
$i^{i^{i}}$
Complex-valued signals
Complex numbers and phasors play a very important role
in electrical engineering. Solving systems for complexexponentials is much easier than for sinusoids, and
linear systems analysis is particularly easy.
Find the phasor representation for each, andre-express each as the real and imaginary parts of a
complex exponential. What is the frequency (in Hz)of each? In general, are your answers unique? If
so, prove it; if not, find an alternative answer forthe complex exponential representation.
$3\sin (24t)$
$\sqrt{2}\cos (2\pi \times 60t+\frac{\pi}{4})$
$2\cos (t+\frac{\pi}{6})+4\sin (t-\frac{\pi}{3})$
Show that for linear systems having real-valued
outputs for real inputs, that when the input is thereal part of a complex exponential, the output is
the real part of the system's output to the complexexponential (see
[link] ).
$$S(\Re (Ae^{i\times 2\pi ft}))=\Re (S(Ae^{i\times 2\pi ft}))$$
For each of the indicated voltages, write it as the real
part of a complex exponential(
$v(t)=\Re (Ve^{st})$ ).Explicitly indicate the value of the complex amplitude
$V$ and the complex frequency
$s$ .
Represent each complex amplitude as a vector in the
$V$ -plane, and indicate the
location of the frequencies in the complex
$s$ -plane.
A particularly interesting communication channel can be
modeled as a linear, time-invariant system. When thetransmitted signal
$x(t)$ is a pulse, the received signal
$r(t)$ is
as shown .
What will be the received signal when the transmitter
sends the
pulse sequence${x}_{1}(t)$ ?
What will be the received signal when the transmitter
sends the
pulse signal${x}_{2}(t)$ that has half the duration as the original?
Analog computers
So-called
analog computers use circuits to solve mathematical problems, particularly when they involve differential equations.
Suppose we are given the following differential equation to solve.
$$\frac{d y(t)}{d t}+ay(t)=x(t)$$ In this equation,
$a$ is a constant.
When the input is a unit step
(
$x(t)=u(t)$ ), the output is given by
$y(t)=(1-e^{-(at)})u(t)$ .
What is the total energy expended by the input?
Instead of a unit step, suppose the input is a unit pulse (unit-amplitude, unit-duration) delivered to the circuit at time
$t=10$ .
What is the output voltage in this case?Sketch the waveform.
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest.
Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.?
How this robot is carried to required site of body cell.?
what will be the carrier material and how can be detected that correct delivery of drug is done
Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?