What is Complex conjugate: Definition and 79 Discussions

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of


{\displaystyle a+bi}
is equal to



{\displaystyle a-bi.}
The complex conjugate of


{\displaystyle z}
is often denoted as


{\displaystyle {\overline {z}}}
In polar form, the conjugate of




{\displaystyle re^{i\varphi }}




{\displaystyle re^{-i\varphi }}
. This can be shown using Euler's formula.
The product of a complex number and its conjugate is a real number:






{\displaystyle a^{2}+b^{2}}



{\displaystyle r^{2}}
in polar coordinates).
If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root.

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  1. D

    P&S Exercise 3.4 Majorana Fermions Derivative of ##\chi##

    I am stuck at the final part where one is supposed to show that the derivative of the second term of the action gives the mass term in the Majorana equation. For $$\chi^T\sigma^2\chi = -(\chi^\dagger\sigma^2\chi^*)^*$$ we get $$\frac{\delta}{\delta\chi^\dagger}(\chi^\dagger\sigma^2\chi^*)^*$$...
  2. Leo Liu

    I Why Does the Complex Conjugate Involve Negating the Argument Theta?

    Can someone please tell me why this is true? This isn't exactly the De Moivre's theorem. Thank you.
  3. docnet

    Complex conjugate of a pole is a pole?

    This isn't a homework problem, but a more general question. Let ##f## be a function with two singular points ##r## and its complex conjugate ##r^*##. let $$f=\frac{g}{z-r} \quad \text{and assume} \quad g(r)\neq 0$$ so ##r## is a simple pole of ##f##. we have conjugates that are singular...
  4. Tony Hau

    I The derivative of the complex conjugate of the wave function

    It is a rather simple question: In my textbook it writes something like: $$\frac {\partial \Psi} {\partial t}= \frac{i\hbar}{2m}\frac {\partial^2 \Psi} {\partial x^2}- \frac{i}{\hbar}V\Psi$$ $$\frac {\partial \Psi^*} {\partial t}= -\frac{i\hbar}{2m}\frac {\partial^2 \Psi^*} {\partial...
  5. I

    I Complex Conjugate of Wave Function

    I've been studying quantum mechanics this semester in school and have ran into an issue I can't find an answer for. I understand why we take the complex conjugate of the wave function, such as when calculating expectation values. I'm a little confused though as to why we take the complex...
  6. G

    What is the complex conjugate of this wave function?

    I was planning to find the value of N by taking the integral of φ*(x)φ(x)dx from -∞ to ∞ = 1. However, this wave function doesn't have a complex number so I'm not sure what φ*(x) is. I was thinking φ*(x) is exactly the same φ(x), but with x+x0 instead of x-x0. Thank you
  7. P

    I Complex conjugate of an inner product

    Hi everyone. Yesterday I had an exam, and I spent half the exam trying to solve this question. Show that ##\left\langle\Psi\left(\vec{r}\right)\right|\hat{p_{y}^{2}}\left|\phi\left(\vec{r}\right)\right\rangle =\left\langle...
  8. T

    I Informational content in 2D discrete Fourier transform

    When you do a discrete Fourier transform (DFT) of a one-dimensional signal, I understand that the second half of the result is the complex conjugate of the first half. If you threw out the second half of the result, you're not actually losing any data and you would be able to recreate the entire...
  9. N

    Unitary transformation using Python

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  10. B

    A Laser beam represented with complex conjugate?

    Boyd - Nonlinear Optics page 5, there says 'Here a laser beam whose electric field strength is represented as $$\widetilde{E}(t) = Ee^{-iwt} + c.c$$But why is it written like this? Is it because the strength is the real part of the complex electric field? Then why doesn't he divide it by 2 after...
  11. Thejas15101998

    I Operation on complex conjugate

    Why do we sandwich operators in quantum mechanics in such a way that the operator acts on the wavefunction and not on its complex conjugate?
  12. Jamz

    I Map from space spanned by 2 complex conjugate vars to R^2

    Hello, I would like your help understanding how to map a region of the space \mathbb{C}^2 spanned by two complex conjugate variables to the real plane \mathbb{R}^2 . Specifically, let us think that we have two complex conugate variables z and \bar{ z} and we define a triangle in the...
  13. J

    I What does complex conjugate of a derivate mean?

    An exercise asks me to determine whether the following operator is Hermitian: {\left( {\frac{d}{{dx}}} \right)^ * }. I don't even know what that expression means. a) Differentiate with respect to x, then take the complex conjugate of the result? b) Take the complex conjugate, then...
  14. redtree

    A Deriving Probability Amplitude from Markov Density Function

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  15. Adolfo Scheidt

    I Product of complex conjugate functions with infinite sums

    Hello there. I'm here to request help with mathematics in respect to a problem of quantum physics. Consider the following function $$ f(\theta) = \sum_{l=0}^{\infty}(2l+1)a_l P_l(cos\theta) , $$ where ##f(\theta)## is a complex function ##P_l(cos\theta)## is the l-th Legendre polynomial and...
  16. dumbdumNotSmart

    Complex Conjugate Inequality Proof

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  17. caters

    How Do You Form a Degree 4 Polynomial with Given Complex Zeros?

    Homework Statement Form a polynomial whose zeros and degree are given below. You don't need to expand it completely but you shouldn't have radical or complex terms. Degree 4: No real zeros, complex zeros of 1+i and 2-3i Homework Equations (-b±√b^2-4ac)/2a The Attempt at a Solution I want...
  18. D

    I Complex conjugate and time reversal operator

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  19. Q

    I Schrodinger equation in terms of complex conjugate

    I know there's a similar post, but i didn't understand it. Why the derivative respect to t in terms of the complex conjugate of ψ is: instead of being the original S.E in terms of ψ* or the equation in terms of ψ with the signs swapped
  20. J

    I What is the complex conjugate of the momentum operator?

    Hello, i am kind of confused about something. What is the complex conjugate of the momentum operator? I don't mean the Hermitian adjoint, because i know that the Hermitian adjoint of the momentum operator is the momentum operator. Thanks!
  21. Oaxaca

    Complex Conjugates with sin and cos

    I am rather new to the whole idea of complex conjugates and especially operators. I was trying to understand the solution to a problem I was doing, but the math is confusing me rather than the physics. In the last row of calculations, why does the sin change to a cos, and the d/dx change to...
  22. Indianspirit

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  23. ognik

    MHB Path dependance of complex conjugate

    Hi, an exercise asks to show that $ \int_{0,0}^{1,1} {z}^{*}\,dz $ depends on the path, using the 2 obvious rectangular paths. So I did: $ \int_{c} {z}^{*}\,dz = \int_{c}(x-iy) \,(dx+idy) = \int_{c}(xdx + ydy) + i\int_{c}(xdy - ydx) = \frac{1}{2}({x}^{2} + {y}^{2}) |_{c} + i(xy - yx)|_{c}...
  24. N

    Repeated complex conjugate roots for Cauchy-Euler

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  25. LachyP

    Calculate Complex Conjugate of Ψ(x,t) for x=4, t=9

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  26. nmsurobert

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  27. KleZMeR

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  28. dkotschessaa

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  29. Muthumanimaran

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  30. B

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  31. F

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  32. S

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  33. J

    What is the dot product of complex conjugate vectors?

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  34. N

    MHB Find Complex Conjugate of 1/(1+e^(ix))

    Good Day, I would like to know how to find the complex conjugate of the complex number 1/(1+e^(ix)). I got (1+e^(-(ix)))/(2+2 cos x) but the solution is 0.5 sec (x/2) e^(i(x/2)). Any help will be greatly appreciated. Thanks & Regards P.S. Apologies for not using LATEX as it was formatting...
  35. A

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  36. S

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  37. N

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  38. I

    Complex conjugate as a Mobius transformation

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  39. J

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  40. T

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  41. H

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  42. S

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  43. B

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  44. S

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  45. C

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  46. S

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  47. S

    Derivative wrt Complex Conjugate

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  48. V

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  49. A

    Complex Conjugate: just replace i by -i even in denominator or inside argument?

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