What is Complex: Definition and 1000 Discussions

The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London (UCL). The Faculty, the UCL Faculty of Engineering Sciences and the UCL Faculty of the Built Envirornment (The Bartlett) together form the UCL School of the Built Environment, Engineering and Mathematical and Physical Sciences.

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  1. G

    I Solving complex formulas and higher order polynomials

    How would you go about solving (4(x^3)+38(x^2)+44x-20)/(20+12x+x^2), without the use of a computer, further, what about functions which have more x components, with higher powers. Also what process do computers use to solve these.
  2. A

    I How can I write a normalized spin superposition in a simplified form?

    If I have an normalized spin superposition |ψ> = (a1+a2i) |1> + (b1+b2i) |2>, and asked to write it in the form |ψ> = cosθ|1> + eiΦsinθ|2>, how do I proceed? My main problem is that no matter what I try, I can't seem to get rid of some complex component that shows up in the coefficient of |1>...
  3. Haley Heldt

    How Do You Calculate Net Force on a Diver?

    Homework Statement An 82 kg man drops from rest on a diving board 3 m above the surface of the water and comes to rest .55 s after reaching the water. What is the net force on the diver as he is brought to rest Homework Equations FDt = mv F = mg v^2 = vi^2 = 2aDx Dx = 3m Dt...
  4. 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...
  5. A

    I Is There a Significance to the Imaginary Number in the Series for Pi/4?

    I find this interesting. You can approximate pi/4 with the Gregory and Leibniz series pi /4 = 1/1 - 1/3 + 1/5 - 1/7 + 1/9 ... (1) btw it takes a lot of terms to get a reasonable approximation for pi. The formuli is pi / 4 = [ ( -1 ) ^ ( k + 1 ) ] / ( 2 * k -1)...
  6. matt_crouch

    How do I solve complex contour integrals in complex analysis?

    I am trying to teach myself complex analysis . There seems to be multiple ways of achieving the same thing and I am unsure on which approach to take, I am also struggling to visualise the problem...Would someone show me step by step how to solve for example...
  7. L

    Geometric Interpretation of complex numbesr

    z1,z2,z3 are distinct complex numbers, prove that they are the vertices of an equilateral triangle if and only if the following relation is satisfied: z1^2+z2^2+z3^2=z1.z2+z2.z3+z3.z1 so i shall show that |z1-z2|=|z1-z3|=|z2-z3|but i do not know how to start.
  8. K

    How to determine the sign of currents in complex circuits?

    Homework Statement A combination circuit powered by a 6.0 V battery is shown. What is the total current through this circuit? I don't know how to determine where the signs go, for example if the right side of a resistor is positive or the left side is positive Homework Equations What is the...
  9. Z

    Differentiating an exponential with a complex exponent

    Hello, folks. I'm trying to figure out how to take the partial derivative of something with a complex exponential, like \frac{\partial}{\partial x} e^{i(\alpha x + \beta t)} But I'm not really sure how to do so. I get that since I'm taking the partial w.r.t. x, I can treat t as a constant term...
  10. B

    How Do I Solve Complex Equations with Different Lambda Values?

    Homework Statement Question 3.b. - http://imgur.com/ztLiRvx Homework Equations For the sake of simplicity, let's assume that lambda = x. The Attempt at a Solution I tried equating the real an imaginary parts of arctan(1/4). Real: x/2 + 3 = 4. This gives x = 2. Imaginary: x/2 - 3 = 1. This...
  11. Spinnor

    Cross between helicoid, complex plane wave

    Function kind of cross between a helicoid and a complex plane wave? I would like to translate a mental picture into a mathematical expression if possible. The picture might be roughly thought of as a cross between a complex plane wave and a helicoid. A construction I think goes as follows, take...
  12. Steve Turchin

    Is this complex vector orthogonal to itself?

    Is the basis vector ##(i,0,1)## in the space ##V=##Span##((i,0,1))## with a standard inner product,over ##\mathbb{C}^3## orthogonal to itself? ##<(i,0,1),(i,0,1)> = i \cdot i + 0 \cdot 0 + 1 \cdot 1 = -1 + 1 = 0 ## The inner product (namely dot product) of this vector with itself is equal to...
  13. U

    Engineering Not sure how to do this complex circuit

    My friend told me the 2 24 are shorted. How do I know these are shorted and b) what do I do do make this question simple?
  14. anorlunda

    Insights The Case for Learning Complex Math - Comments

    anorlunda submitted a new PF Insights post The Case for Learning Complex Math Continue reading the Original PF Insights Post.
  15. O

    Finding Complex Antiderivatives | Guidance for Tricky Functions

    Homework Statement How would one go about finding the antiderivative to this function? Homework Equations N/A The Attempt at a Solution This problem has been rather tricky I have tried several attempts at the solution. My one solution consists of me factoring out the x^4. Looking for some...
  16. D

    How to analyze a complex circuit

    Homework Statement Homework Equations Kirchoff's Current and Voltage laws The Attempt at a Solution How do you go about analyzing a complex circuit like this? Do you just write out the current equation for each highlighted junction and voltage equations for each loop? Is there a quick way...
  17. Isaac0427

    Insights Complex and Irrational Exponents for the Layman - Comments

    Isaac0427 submitted a new PF Insights post Complex and Irrational Exponents for the Layman Continue reading the Original PF Insights Post.
  18. King_Silver

    Complex Number Equations: Solving for z and Finding the Perpendicular Bisector

    Homework Statement a) Solve equation z + 2i z(with a line above it i.e. complex conjugate) = -9 +2i I want it in the form x + iy and I am solving for z. b) The equation |z-9+9i| = |z-6+3i| describes the straight line in the complex plane that is the perpendicular bisector of the line segment...
  19. SrVishi

    Analysis What are some alternatives to Rudin for learning complex analysis?

    Hello, I was wondering how well is Rudin's Real and Complex Analysis for learning complex analysis, assuming that difficulty won't be an issue. Does it cover the standard material? Is it deep enough? Should I just read from elsewhere instead?
  20. C

    Schools Is Complex Analysis a must for grad school applications?

    Is taking complex analysis before graduate school apps a "make-or-break" deal if one is looking to apply for theory? I am currently deciding whether to take it junior spring or defer it to senior spring. As it has come up in my research, I have studied some of it, but I'm wondering if it must be...
  21. anemone

    MHB Evaluating Complex Equation without a Calculator

    Without the help of calculator, evaluate \frac{(-\sqrt{6}+\sqrt{7}+\sqrt{8})^4}{4(\sqrt{7}-\sqrt{6})(\sqrt{8}-\sqrt{6})}+\frac{(\sqrt{6}-\sqrt{7}+\sqrt{8})^4}{4(\sqrt{6}-\sqrt{7})(\sqrt{8}-\sqrt{7})}+\frac{(\sqrt{6}+\sqrt{7}-\sqrt{8})^4}{4(\sqrt{6}-\sqrt{8})(\sqrt{7}-\sqrt{8})}.
  22. TheMathNoob

    Relation between complex eigenvalues and rotations

    Homework Statement I have the following matrix: 0 0 0 1 1 0 0 0 = A 0 1 0 0 0 0 1 0 and the vector v = (1,0,0,0) If I perform Av, this gives: Av=(0,1,0,0) And If I keep multiplying the result by A like A*A*(Av), the outcome will be something like j= (0,0,1,0) k=(0,0,0,1) l=(1,0,0,0) The...
  23. naima

    Fourier transform with complex variables

    I found this formula in a paper: \int exp( \frac{x1 + i x2}{ \sqrt 2} \eta^* - \frac{x1 - i x2}{ \sqrt 2} \eta) D(\eta)/ \pi d^2 \eta the author calls it the Fourier transform of D. It is the first time thar i see this formula. How common is this notation? Can we use it without problem?
  24. M

    Complex numbers simplification

    Homework Statement [/B] Z=((2z1)+(4z2))/(z1)(z2) where Z1=4e^2pi/3 Z2=2/60 degre, z3=1+i The answer must be in polar form r/theta Homework Equations Well in the upper section The Attempt at a Solution After do some operations i get to this and unable to convert to polar form... -...
  25. C

    Can an orthogonal matrix be complex?

    Can an orthogonal matrix involve complex/imaginary values?
  26. M

    Complex derivative and div/curl

    In trying to get an intuition for curl and divergence, I've understood that in the case of R2, div f(x,y) = 2Re( d/dz f(z,z_)) and curl f(x,y) = 2Im( d/dz f(z,z_)), where f(z,z_) is just f(x,y) expressed in z and z conjugate (z_). Is there any way of proving the fundamental properties of div and...
  27. M

    Using complex description of div and curl in 2d?

    In trying to get an intuition for curl and divergence, I've understood that in the case of R2, div f(x,y) = 2Re( d/dz f(z,z_)) and curl f(x,y) = 2Im( d/dz f(z,z)), where f(z,z) is just f(x,y) expressed in z and z conjugate (z). Is there any way of proving the fundamental properties of div and...
  28. micromass

    Insights Things Which Can Go Wrong with Complex Numbers - Comments

    micromass submitted a new PF Insights post Things Which Can Go Wrong with Complex Numbers Continue reading the Original PF Insights Post.
  29. F

    What is the meaning of ph(z) in complex numbers?

    Exactly as stated in the title. What does ph(z) mean?
  30. M

    Finding Residue of Complex Function at Infinity

    Hello everyone, I have a problem with finding a residue of a function: f(z)={\frac{z^3*exp(1/z)}{(1+z)}} in infinity. I tried to present it in Laurent series: \frac{z^3}{1+z} sum_{n=0}^\infty\frac{1}{n!z^n} I know that residue will be equal to coefficient a_{-1}, but i don't know how to find it.
  31. G

    MHB Complex numbers simplification

    If $z = e^{(2-\frac{i \pi}{4})}$ what's $z^5$? The only way I can think of doing this is expanding $(2-\frac{i \pi}{4})^5$, but I think I'm supposed to use a simpler method (not sure what it's).
  32. G

    MHB What is the ratio of complex numbers in the form of a question?

    What's the ratio $\displaystyle \frac{e^{i\sqrt{x}}-1}{e^{i\sqrt{x}}+1}$ equal to? I can't work it out to anything I recognize. :confused: The answer is $\displaystyle i\tan(\frac{1}{2}\sqrt{x})$. I suppose I could work backwards from the answer, but I won't have the answer in the exam.
  33. G

    MHB Quickest way to calculate argument of a complex number

    What's the quickest way to calculate the argument of $\displaystyle \pi e^{-\frac{3i\pi}{2}}$?
  34. ognik

    MHB Please check this complex integral (#2)

    An old exam question is: Evaluate $ \oint \frac{e^{iz}}{z^3}dz $ where the contour is a square of sides a, centered at 0. This has a simple pole of order 3 at z = 0 Perhaps using residues, $ Res(f,0) = \frac{1}{2!}\lim_{{z}\to{0}}\d{^2{}}{{z}^2}z^2 \frac{e^{iz}}{z^3} =...
  35. ognik

    MHB Please check this complex integral

    An old exam has: Evaluate $ \oint\frac{dz}{z(2z+1)} $, where the contour is a unit circle This look good for the residue theorem, it has 2 simple poles at 0, $-\frac{1}{2}$ $ Res(f, 0)= \lim_{{z}\to{0}}z\frac{1}{z(2z+1)}=1$ $ Res(f, -\frac{1}{2})=...
  36. Hijaz Aslam

    Euler Representation of complex numbers

    I am bit confused with the Eueler representation of Complex Numbers. For instance, we say that e^{i\pi}=cos(\pi)+isin(\pi)=-1+i0=-1. The derivation of e^{i\theta}=cos(\theta)+isin(\theta) is carried out using the Taylor series. I quite understand how ##e^{i\pi}## turns out to be ##-1## using...
  37. Einstein's Cat

    Multiplying and dividing real and complex numbers

    Is it possible to divide and multiply complex numbers and real numbers and if so, how does one do that? If not, why so? Cheers for your help!
  38. L

    Complex phase space coordinates

    First post ! I hope that my question will not make some long time physicists laugh. It is about geometrical quantization and the phase space in which we use : z=1/sqrt(2)(q+ip) My question is simple what is this 1/sqrt(2) ? And what is it is interpretation ? Thank you !
  39. H

    Troubleshooting Complex Number Formulas in Matlab

    One problem I sometimes encounter is with complex numbers. When a formula including functions of complex variables runs in Matlab, I obtain the corresponding result but if I write that formula in different forms (for example when I arrange the long formula in simpler form) I obtain another...
  40. Z

    Hydrogen atom ground state wave function complex conjugate

    For hydrogen atom ground state we know φ=π-1/2a-3/2e-r/1 I want to know the complex conjugate of φ* ?
  41. Ricky_15

    Argument of a random complex no. lying on given line segment.

    Homework Statement In the argand plane z lies on the line segment joining # z_1 = -3 + 5i # and # z_2 = -5 - 3i # . Find the most suitable answer from the following options . A) -3∏/4 B) ∏/4 C) 5∏/6 D) ∏/6 2. MY ATTEMPT AT THE SOLUTION We get two points ( -3 , 5 ) & ( -5 , -3 ) => The...
  42. toforfiltum

    Inequalities of negative arguments in complex numbers

    Homework Statement Arg z≤ -π /4 Homework EquationsThe Attempt at a Solution I'm confused whether the answer to that would be more than -45° or less. Should the approach to arguments be the same as in negative numbers?
  43. C

    Finding complex number with the lowest argument.

    Homework Statement Of all complex numbers that fit requirement: ## |z-25i| \leq 15## find the one with the lowest argument. Homework EquationsThe Attempt at a Solution z=a + ib (a, b are real numbers) ## \sqrt{a^2 + (b-25)^2} \leq 15 \\ a^2 + (b-25)^2 \leq 225 ## The lowest possible...
  44. S

    Partial derivative of a complex number

    Homework Statement Given n=(x + iy)/2½L and n*=(x - iy)/2½L Show that ∂/∂n = L(∂/∂x - i ∂/∂y)/2½ and ∂/∂n = L(∂/∂x + i ∂/∂y)/2½ Homework Equations ∂n Ξ ∂/∂n, ∂x Ξ ∂/∂x, as well as y. The Attempt at a Solution ∂n=(∂x + i ∂y)/2½L Apply complex conjugate on right side, ∂n=[(∂x + i ∂y)/2½L] *...
  45. C

    Complex Solution to an Exponential Equation

    Homework Statement Solve the following equation: ## (1+a)^n=(1-a)^n## where a is complex number and n is natural number Homework Equations Euler's formula The Attempt at a Solution I've tried something like this ## (1+a)^n=(1-a)^n \\ (\frac{1+a}{1-a})^n=1 ## But i really have no idea...
  46. B

    Derivative of complex equation

    hello! 1) what is the process to get the derivative of an equation that requires you to do first the chain rule and then the product/quotient rule, eg. sin(x^2(x+1))? 2) what is the process to get the derivative of an equation that requires you to do first the product/quotient rule and then the...
  47. squelch

    Complex Numbers and Constants of Integration

    Homework Statement Suppose that the characteristic equation to a second order, linear, homogeneous differential equation with constant coefficients yielded two complex roots: \begin{array}{l} {\lambda _1} = a + bi\\ {\lambda _2} = a - bi \end{array} This would yield a general solution of: y =...
  48. RicardoMP

    Fixed point method for nonlinear systems - complex roots

    Homework Statement I've been asked to graphically verify that the system of equations F (that I've uploaded) has exactly 4 roots. And so I did, using the ContourPlot function in Mathematica and also calculated them using FindRoot. Now, I've to approximate the zeros of F using the fixed point...
  49. C

    Finding Product of Complex Polynomial Roots

    Homework Statement It is known that roots of complex polynomial: ##P_n (z) = z^n + a_{n-1}z^{n-1} + \cdots + a_1z + a_0## are the following complex numbers: ##\alpha_1, \alpha_2, \cdots, \alpha_n ## Find the product: ##\prod = (\alpha_1 + 1)(\alpha_2 + 1)\cdots(\alpha_n +1)## Homework...
  50. L

    Determining the complex expression using Thevnin's theorem

    I tried my best but I wasn't able to solve this can someone please provide me with a detailed solution. Here 's the question : Establish the expression of Vs/Ve (complex) using Thevnin's theorem Here is the circuit : I spent 4 hours trying to solve this but I had no clue how. I'am having...
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