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:






{\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.

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  1. H

    I Sum of the dot product of complex vectors

    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}##...
  2. BWV

    Worth learning complex exponential trig derivations in precalc?

    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...
  3. H

    Question about the argument in a Complex Exponential

    I know that e^-ix = cos(-x)-isin(x), but if we have e^-iwx does that equal cos(-wx) - isin(wx)? Thanks
  4. K

    I Why the integral of a complex exponential can't be equal to zero?

    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##...
  5. S

    How to write the complex exponential in terms of sine/cosine?

    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)...
  6. Mr Davis 97

    I Complex exponential to a power

    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...
  7. CalcExplorer

    A How to Convert a Complex Logarithm to a Complex Exponential

    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...
  8. nomadreid

    B Mistake in my complex exponentiation: where?

    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...
  9. D

    MHB Complex Exponential Function

    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?
  10. F

    Simplification of a complex exponential

    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...
  11. lahanadar

    Period of a complex exponential signal

    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...
  12. G

    A A problem about branch cut in contour integral

    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...
  13. A

    I Integral with complex oscillating phase

    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\}...
  14. D

    I Complex Exponential solutions in time invariant systems

    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...
  15. P

    Understanding the Significance of Complex Exponential in Electrical Engineering

    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...
  16. wirefree

    Understanding Complex Exponential Summation: How is the Arctan Function Used?

    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
  17. Summer95

    Derivative of complex exponential differs by a sign

    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.
  18. Greg

    MHB How can the complex exponential product be proven for all real p and m?

    Show that, for all real p and m, e^{2mi\cot^{-1}(p)}\left(\dfrac{pi+1}{pi-1}\right)^m=1
  19. C

    MATLAB Transforming Complex Exponential to Discrete Vector Form

    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
  20. L

    Fortran Can complex numbers be used in Fortran array constructors?

    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...
  21. B

    Verifying an Inequality Involving the Complex Exponential Function

    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...
  22. bibo_dvd

    Periodic Complex exponential signal

    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...
  23. L

    Complex exponential equation

    How can we show that the following equation has infinitely many solutions e^z-z^2=0. Thanks
  24. S

    Differentiating complex exponential

    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...
  25. M

    Help with Triangle Wave using complex exponential Fourier Series

    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...
  26. W

    Limit of a complex exponential

    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...
  27. M

    Expressing sum of sines and cosines as a complex exponential

    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...
  28. B

    Integrating power of a cosine times a complex exponential

    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...
  29. J

    Equation with two unknowns in complex exponential

    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 *...
  30. T

    Integration via complex exponential

    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...
  31. M

    Finding the magnitude of a complex exponential function

    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...
  32. G

    Absolute value of complex exponential equals 1

    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...
  33. S

    Complex exponential and sine-cosine Fourier series

    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...
  34. M

    Complex exponential to trigonometric simplification

    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...
  35. K

    URGEN taking the power of complex exponential

    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
  36. L

    Complex exponential function

    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...
  37. S

    Integrating the product of a real and a complex exponential

    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...
  38. B

    MHB Quick modulus question - complex exponential function

    What is |exp (z^n)| less than if |z| < 1? I'm thinking it's e but I'm having a brain freeze at the moment! Thanks for any help guys.
  39. F

    Help needed on complex exponential equation

    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 :)
  40. A

    Trigonometric function and complex exponential

    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
  41. J

    [Complex exponential] Solutions must be wrong

    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...
  42. L

    Complex exponential expressions.

    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...
  43. S

    Complex exponential description of SHM

    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...
  44. S

    Solve complex exponential equation

    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...
  45. F

    Fourier transform, complex exponential and infinity

    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...
  46. P

    Complex exponential X delta function

    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...
  47. E

    Magnitude of Complex Exponential Polynomial Inequality

    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...
  48. F

    The CT complex exponential is NOT periodic

    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...
  49. A

    Square root of negative complex exponential

    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...