What is Complex exponential: Definition and 77 Discussions
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x:
e
i
x
=
cos
x
+
i
sin
x
,
{\displaystyle e^{ix}=\cos x+i\sin x,}
where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively. This complex exponential function is sometimes denoted cis x ("cosine plus i sine"). The formula is still valid if x is a complex number, and so some authors refer to the more general complex version as Euler's formula.Euler's formula is ubiquitous in mathematics, physics, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics".When x = π, Euler's formula evaluates to eiπ + 1 = 0, which is known as Euler's identity.
Summary:: summation of the components of a complex vector
Hi,
In my textbook I have
##\widetilde{\vec{E_t}} = (\widetilde{\vec{E_i}} \cdot \hat{e_p}) \hat{e_p}##
##\widetilde{\vec{E_t}} = \sum_j( (\widetilde{\vec{E_{ij}}} \cdot {e_{p_j}}*) \hat{e_p}##
For ##\hat{e_p} = \hat{x}##...
This is a pedagogical /time management / bandwidth / tradeoff question, no argument that learning the complex exponential derivation is valuable, but is it a good strategy for preparing for first year Calculus? my 16YO son is taking AP precalc and AP calc next year and doing well, but struggled...
I was just playing with the integral ##\int e^{ixa}dx## when I found something interesting. If you integrate from ##x = m2\pi/a## to ##x = n2\pi/a## where ##m## and ##n## are any two integers, the integral equals zero.
On one hand, as we can in principle choose whatever values we like for ##m##...
I apologize in advance if any formatting is weird; this is my first time posting. If I am breaking any rules with the formatting or if I am not providing enough detail or if I am in the wrong sub-forum, please let me know.
1. Homework Statement
Using Euler's formula : ejx = cos(x) + jsin(x)...
Say I have ##e^{2\pi i n}##, where ##n## is an integer. Then it's clear that ##(e^{2\pi i})^n = 1^n = 1##.
However, what if replace ##n## with a rational number ##r##? It seems that by the same reasoning we should have that ##e^{2\pi i r} = (e^{2\pi i})^r = 1^r = 1##. But what if ##r=1/2## for...
Okay, so I'm working with a rather frustrating problem with a calculus equation. I'm trying to solve a calculus equation which I conceptualized from existing methods involving complex number fractal equations. I'm very familiar with pre-calculus, while being self-taught in portions of calculus...
I am sure I am overlooking something elementary, but playing around with exponentiation (this is not an assignment), I seem to be making a mistake somewhere. Please don't send me a link for a more compact way of getting the correct result; I wish to know what my particular mistake is.
Suppose...
Let f(z) = $e^{e^{z}}$ . Find Re(f) and Im(f).
I don't know how to deal with the exponential within an exponential. Does anybody know how to deal with this?
Homework Statement
Is there a way to simplify the following expression?
##[cos(\frac {n \pi} 2) - j sin(\frac {n \pi} 2)] + [cos(\frac {3n \pi} 2) - j sin(\frac {3n \pi} 2)]##
Homework Equations
##e^{jx} = cos(x) + j sin(x)##
The Attempt at a Solution
##cos(\frac {n \pi} 2)## and...
I have a simple complex exponential signal of the form x(t)=ejωt. To find period of the signal I tested if x(t)=x(t+nT) for all n:
ejωt=ejω(t+nT) ⇒ ejωnT=1=ej2πk
where n and k are integers. Then I find a general period expression as
T=2πk/ωn
Period T means it is the least time a signal...
Hello.
I have a difficulty to understand the branch cut introduced to solve this integral.
\int_{ - \infty }^\infty {dp\left[ {p{e^{ip\left| x \right|}}{e^{ - it\sqrt {{p^2} + {m^2}} }}} \right]}
here p is a magnitude of the 3-dimensional momentum of a particle, x and t are space and time...
Does there exist and analytical expression for the following integral?
I\left(s,m_{1},m_{2},L\right)=\sum_{\boldsymbol{n}\in\mathbb{N}^{3}\backslash\left\{ \boldsymbol{0}\right\}...
Hi there! First Post :D
In a recent CM module we've been looking at coupled oscillators and the role of time translational invariance in the description of such physical systems. I will present the statement that I am having trouble understanding and then continue to elaborate.
In stating that...
In my electrical engineering textbook, they have an entire chapter devoted to the complex Exponential. I don't really understand it, nor do I understand its importance.
I know it is extremely important, and need to understand why, and what exactly it is, and the wording of the online resources...
I appreciate the opportunity afforded by this forum to submit a question.
I have struggled with the derivation shown in the attached picture. I am certainly unfamiliar with the concept used to include the arctan function in the encircled step.
Would be highly appreciative of a prompt.wirefree
I know this is probably the least of my worries at the moment but my quantum textbook solves ##\frac{\mathrm{d}\phi (t) }{\mathrm{d} t}=\frac{iC}{h}\phi (t) ## as ##\phi (t) = e^{-i(\frac{C}{h})t}##. Is this not off by a sign? Its really bugging me.
Hi,
I want to transform a complex exponential with quadratic phase to discrete form, in other words to a vector form.
can anyone help me with that?
Thanks
Hi, so I need to write a fortran code with 2, 2x2 matrices.
These matrices are in the form of B=(1 exp(i)(theta) 0 0) and D=(0 0 exp(i)(theta) 1) where i is sqrt of -1 and theta is an angle between 0 and 2pi.
I've expanded the exponential so it reads cos(theta)+isin(theta) and let theta=pi/2...
Demonstrate that ##|e^{z^2}| \le e^{|z|^2}##
We have at our disposal the theorem which states ##Re(z) \le |z|##. Here is my work:
##e^{|z|^2} \ge e^{(Re(z))^2} \iff## By the theorem stated above.
##e^{|z|^2} \ge e^x##
We note that ##y^2 \ge 0##, and that multiplying by ##-1## will give us...
hello guys ..
first of all , iam not sure that i should type this thread here . so excuse me for that
in this problem i can understand the part until it's said that w=0 then x(t)=1, which is periodic for any value of T
but i can't understand the part after that in the case of w is not equal...
I asked to differentiate the given function using exponential function
with sin(√3t + 1) I turned it into Im[e^(√3t+1)i]
then I multiplied it by e^t
which gave Im[e^t*e^(√3t +1)i]
then I applied usual algebra to differentiate but I get a (t+√3ti +i) as the power of e
when I try...
I'm participating in research this summer and it's has to do with the Fourier Series. My professor wanted to give me practice problems before I actually started on the research. He gave me a square wave and I solved that one without many problems, but this triangle wave is another story. I've...
Hi,
The two terms should vanish at infinity according to the Quantum textbook of Griffiths, but I don't see how?
I mean a complex exponential is a periodic function so how can it vanish at infinity?
If you split up the first term
exp(ikx) * exp(-ax)
Take the limit of infinity...
If I'm given a function ##f(x) = A cos (x) + B sin (x)##, is there any way to turn this into an expression of the form ##F(x) = C e^{i(x + \phi)}##? I know how to use Euler's formula to turn this into ## \alpha e^{i(x + \phi)} + \beta e^{-i(x + \phi)}##, but is there a way to incorporate the...
Homework Statement
Consider the inner product
$$\frac{1}{2\pi}\int_0^{2\pi} \left(\frac{3}{5 - 4\cos(x)}\right) e^{-ikx} dx, \quad k \in \mathbb{Z}, \quad x \in \mathbb{R}.$$
Homework Equations
Is there a method to solve this without using the residue theorem, e.g. integration by parts...
hello friends,
when i build the mathmatical model of robot,i face a new question that i ever seen before.
i have a reverse kinematic lever as the leg and i want to use the tip position to get the relationship of fold angle and rotate angle reversely
here is my equation:
x*e^iθ - y*e^iθ *...
Homework Statement
Using the complex exponential, nd the most general function f such that
\frac{d^2f}{dt^2} = e-3t cos 2t , t all real numbers.
Homework Equations
I'm having a lot of trouble with this question, my thinking is to integrate once and then one more time...
Homework Statement
I want to know the steps involved in finding the magnitude of a complex exponential function. An example of the following is shown in this picture:
Homework Equations
|a+jb|=sqrt(a^2+b^2)
|x/y|=|x|/|y|
The Attempt at a Solution
For the denominator, I replaced z with e^jw...
Hello all,
I'm having trouble showing that |e^it|=1, where i is the imaginary unit. I expanded this to |cos(t)+isin(t)| and then used the definition of the absolute value to square the inside and take the square root, but I keep getting stuck with √(cos(2t)+sin(2t)). Does anyone have any...
The sine-cosine (SC) Fourier series: $$f(x) = \frac{A_0}{2} + \sum_{j=1}^{+\infty} A_j cos(jx) + \sum_{j=1}^{+\infty} B_jsin(jx) $$
This form can also be expanded into a complex exponential (CE) Fourier series of the form: $$ f(x) = \sum_{n=-\infty}^{+\infty} C_n e^{inx} $$
and vice versa...
Homework Statement
Given (e^(ix) - 1)^2 , show that it is equal to 2-2cosx
Homework Equations
e^ix = cosx + isinx
The Attempt at a Solution
After subbing in Euler identity and expanding I get:
cos(x)^2+sin(x)^2-2cosx-2jsinx+2jcosxsinx + 1
after using the addtion formulas I get...
Homework Statement
e^(i*2pi*1/15) is equal to ( e^(i*2pi) )^(1/15) = (1)^(1/15)=1
Why this is false?
Homework Equations
((A)^(b))^c=A^(b*c)=A^(bc). Why this isn't the case for complex exponential?
The Attempt at a Solution
Homework Statement
Reading Hinch's book, there is a statement as follows:
... z need to be kept in the sector where exp(-z^2) ->0 as z -> infinity. Thus it's applicable to the sector |arg z|<pi/4...Homework Equations
Why is this true and what is the limiting behavior of exp(x) for x in...
Homework Statement
\Psi(x,t) = \int^{\infty}_{-\infty} C(p)\Psi_{p}(x,t) dp
is a solution to the Schroedinger equation for a free particle, where
\Psi_{p}(x,t) = Ae^{i(px-Ept)/\hbar}.
For the case C(p) = e^{-(p-p_{0})^{2}/\sigma}
where \sigma is a real constant, compute the wavefunction...
Hi
I have come across this equation:
zn-1=\bar{z}, z\inℂ* (ℂ*:=ℂ\0ℂ)
There are numerous obvious equalities that can be used, but I don't seem to reach a satisfying final answer.
Any help would be appriciated.
Thanks in advance :)
1. Homework Statement
- multiplication of trigonometric function and complex exponential
2. Homework Equations
the question is, Akcos(ωt) × e-jωt
3. The Attempt at a Solution
it is, Ak/2 + (Ak/2)e-j2ωt ?
by using cos(ωt) = 1/2ejωt + 1/2e-jωt
Homework Statement
Hi,
I would like to get feedback if my z-plot is accurate for the following complex exponentiala:
a=2*exp(j*∏*t)
b=2*exp(j*∏*-1.25)
c=1*exp(j*∏*t)
d=-j*exp(j*∏*t)
Further analysis:
a= -2 because cos(∏) = -1 and sin(∏)=0
b= actual complex number A*cos(∅)+j*sin(∅)
c= -1...
Homework Statement
I just need some kind of explanation in layman's terms of what exactly is going on here. It seems as though I am missing some key element from trig. I am in a Signals class and the book lacks an explanation of the reduction used and ultimately why.
Homework...
Hey,
I'm currently reading a textbook which is attempting to derive the equation for a standing wave from first principles. I understand most steps with the exception of one.
It derives a sinusoidal function {x = A \sin \omega t} from a second order ODE, but then immediately interchanges...
I'm having some trouble solving for t in the following exponential equation.
$$ B = A_1 e^{-\lambda_1 t} + A_2 e^{-\lambda_2 t} $$
I can't divide out the leading coefficients A1 and A2 because they differ. I'm not really sure how to immediately take the natural logarithm of both sides...
I'm taking the Fourier transform of a signal. This integral has bounds from -∞ to ∞, but since the signal is 0 for negative t, the bounds become 0 to ∞
doing the integration, the antiderivative I get is et*(-3-jω+2j) where j is sqrt(-1)
Now I have to evaluate this at t=infinity (since it is a...
1. Problem Statment:
Sketch the sequence x(n)=\delta(n) + exp(j\theta)\delta(n-1) + exp(j2\theta)\delta(n-2) + ...
3. Attempt at the Solution:
The angle theta is given in this case Can someone remind me of how to multiply a complex exponential by a delta function? This sequence represents...
Homework Statement
Digital filter analysis - this is just one part of a multi-part question I can't move forward with. It's supposed to be an auxilliary question and isn't the "meat" of the problem.
Find b, such that maximum of the magnitude of the frequency response function...
I'm taking a signals and systems class and the textbook (Signals and systems by Oppenheim) says the CT complex exponential of the form x(t) = C eat with C and a complex is a periodic signal. I fail to see how.
Let C = |C| ejα (exponential form of a complex number)
and a = r + jω (rectangular...
Homework Statement
Solve \sqrt{-e^{(i2\pi)/3}}
Homework Equations
The Attempt at a Solution
I seem to be missing something simple, as I take:
\sqrt{-1} = i
then,
e^{(1/2)*(i2\pi)/3}
which comes out as: ie^{i\pi/3}
however, the solution is:
-ie^{i\pi/3}, and I can't seem to see where...