What is Complex equation: Definition and 68 Discussions

A complex differential equation is a differential equation whose solutions are functions of a complex variable.
Constructing integrals involves choice of what path to take, which means singularities and branch points of the equation need to be studied. Analytic continuation is used to generate new solutions and this means topological considerations such as monodromy, coverings and connectedness are to be taken into account.
Existence and uniqueness theorems involve the use of majorants and minorants.
Study of rational second order ODEs in the complex plane led to the discovery of new transcendental special functions, which are now known as Painlevé transcendents.
Nevanlinna theory can be used to study complex differential equations. This leads to extensions of Malmquist's theorem.

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  1. V

    Finding Values of Complex Equation

    I took the equation and rewrote it as: e(i+1)(log(1-i)-log(√2)). So I worked on it in sections meaning e(i+1) and then log(1-i). For e(i+1) I got eie1 and used Euler's formula for ei to get: e1(cos(1)+isin(1)). And then for log(1-i) I got ln√2 + i(-(π/4)+2kπ). Do I just bring them...
  2. M

    Mathematica Find the real zeros of a complex equation

    Hi PF! Here's my equation 0.5 Sech[0.997091 Sqrt[ 2.4674 + \[Xi]\[Xi]^2]] (-1. ((0. + 3.7011 I) + (0. + 1.5 I) \[Xi]\[Xi]^2) Cosh[ 0.997091 Sqrt[2.4674 + \[Xi]\[Xi]^2]] - 1. \[Sqrt](((0. + 3.7011 I) + (0. + 1.5 I) \[Xi]\[Xi]^2)^2 Cosh[ 0.997091 Sqrt[2.4674 +...
  3. L

    Solving a complex equation (damping/exponential-decay) like this....?

    ##e^{-0.6x}\sin{(5x)}-0.1=0## I have posted my graphical solution to this problem. But, how do I solve this numerically/mathematically without graphing it?
  4. Adesh

    How to solve a complex equation to get the current?

    I was reading The Feynman Lectures on physics http://www.feynmanlectures.caltech.edu/I_23.html chapter 23, section 4. In it he derives the equation for current when inductor, resistor and capacitor is connected in series with an alternating voltage source, he derives this equation:-...
  5. K

    Solving Complex Equations: z2+2(1-i)z+7i=0

    Homework Statement So it is pretty straight forward, solve this. z2+2(1-i)z+7i=0 Homework Equations z2+2(1-i)z+7i=0 (-b±√(b2-4ac))/2a The Attempt at a Solution So what I would do first is solve 2(2-1)z, I get (2-2i)z=2z-2iz we now have z2-2iz+7i+2z=0 Now I don't really know what to do because...
  6. I

    I Drude Model Permittivity Formula - e^iwt or e^(-iwt)?

    I've been having a sign problem while deriving the permittivity formula using Drude model, and I found out that the problem came from the fact that complex field vectors are expressed with e-iwt, not eiwt, thus producing (-iwt) term when differentiated...
  7. N

    Sinusoids as Phasors, Complex Exp, I&Q and Polar form

    Hi, I am going around in circles, excuse the pun, with phasors, complex exponentials, I&Q and polar form... 1. A cos (ωt+Φ) = Acos(Φ) cos(ωt) - Asin(Φ)sin(ωt) Right hand side is polar form ... left hand side is in cartesian (rectangular) form via a trignometric identity? 2. But then...
  8. C

    MHB Stuck solving a complex equation for T

    Hi I wonder if anyone can help. I am not even sure I am on the right forum. I cannot solve this equation for t. It is the final sequence of a number of equations in a book about modelling athletic performance using bioenergetics. I had a high school maths education 40 years ago and I’m stumped...
  9. Rectifier

    Solving Complex Equation: $$ \bar{z} = z^n $$

    The problem I would like to solve: $$ \bar{z} = z^n $$ where ##n## is a positive integer. The attempt ## z = r e^{i \theta} \\ \\ \overline{ r e^{i \theta} } = r^n e^{i \theta n} \\ r e^{-i \theta} = r^n e^{i \theta n} ## ## r = r^n \Leftrightarrow true \ \ if \ \ n=1 \ \ or \ \ r=1## ##...
  10. Y

    MHB How to Avoid Extraneous Solutions in Solving Complex Equations

    Hello all, Please look at the following: Solve the equation: \[\left | z \right |i+2z=\sqrt{3}\] where z is a complex number. I tried solving it, and did the following, which is for some reason wrong. I saw a correct solution. My question to you is why mine is not, i.e., where is my mistake...
  11. W

    Geometric interpretation of complex equation

    Homework Statement $$z^2 + z|z| + |z|^2=0$$ The locus of ##z## represents- a) Circle b) Ellipse c) Pair of Straight Lines d) None of these Homework Equations ##z\bar{z} = |z|^2## The Attempt at a Solution Let ##z = r(cosx + isinx)## Using this in the given equation ##r^2(cos2x + isin2x) +...
  12. Alettix

    What Are the Loci for Different Values of Lambda in a Complex Equation?

    Homework Statement Consider the relation ## |\frac{z-i}{z*-i}| = \lambda ## where z = x + yi a) For ##\lambda = 1## show that the locus is a line in the complex plane and find its equation. b) What is the locus when ##\lambda = 0##? c) Show that for all other positive ##\lambda## the locus may...
  13. 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})}.
  14. 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...
  15. karush

    MHB What Are the Solutions to the Equation \(x^2 + 2i = 0\)?

    $${x}^{2}+2i=0$$ $$\left(x-? \right)\left(x-? \right)=0$$ This should be easy but I couldn't get the factor
  16. Albert1

    MHB Find the Minimum Value of a Complex Equation

    $a>1,b>1$ find the minimum value of $\dfrac {a^2}{b-1}+\dfrac {b^2}{a-1}$
  17. T

    Finding root of complex equation

    Homework Statement Good day, I've been have having difficulties finding the roots of this: Find the roots of 3ix^2 + 6x - i = 0 where i = complex number i = sqrt(-1) Homework Equations quadratic formula (apologies for the large image) The Attempt at a Solution...
  18. gimak

    The second one is a PDF file so you will need Adobe Acrobat Reader to read it.

    Homework Statement y=[(1+y^2)^1.5]/[2(y+sqrt 3)^2]; solve for y Homework Equations see above The Attempt at a Solution I tried to use algebra to solve it, but I can't. The textbook says it can be solved numerically or by iteration. By numberically I think it means algebraically. But I don't...
  19. F

    MHB What is the missing term in equation (E)?

    Good evening everyone, here is the statement: Note the equation (E): Z³-12z² + = 48z-128. 1.Check that 8 is the solution of (E) 2.a) determine real a, b, c such that for all z C (overall) Z³ 12z²-48z-128 + = (z-8) (az² + bz + c). b) Solve the equation in C (E)...
  20. J

    Algorithm to partition a complex equation into 2 master equations

    My Calculus tool is coming along. The only thing left is to write some it's helper functions, such as the one described below: void CalculusWizard::partitionEquation(const std::string & eq, std::string & eq1, std::string & eq2, CalcWizConsts::eqOps & oper) { /* Given an equation eq, partion...
  21. Z

    How can I solve a complex equation without a good calculator?

    15=343*sin*x(0.357143 (+1.41421 sqrt(2389.-2401. cos(2. x))+98. sin(x))-4.9((0.357143 (+1.41421 sqrt(2389.-2401. cos(2. x))+98. sin(x)))^2 I do not have a good enough calculator please help solve for x. Thank you
  22. Q

    Trying to solve complex equation with 1 variable,

    Homework Statement Guys, could you please help me solve this equation for x: 220000=\sqrt{1.4*10^{32}*\left[ln(\frac{6.17*10^{20}+x}{6.17*10^{20}})-\frac{x}{6.17*10^{20}+x}\right]*\frac{1}{x}} It's supposed to give me the radius of the Milky Way's dark matter halo. I expect to get a...
  23. A

    Find all values complex equation

    Homework Statement i.5 - i Homework Equations zc=ec log(z) z1/n=exp[(1/n) log(z)], n is in integer The Attempt at a Solution letting c=.5-i and z=i so zc=e(.5-i) log(i) = e.5 log(i)*e-i log(i) from the second equation I reduced it to: i.5*e-i log(i) but I'm not sure...
  24. B

    Finding all pairs of values that satisfy complex equation

    Homework Statement Find all pairs of values a and b that satisfy (a + bi)2 = 48 + 14i 2. The attempt at a solution I managed to solve it, but it took a while and I was wondering if there is an easier/quicker way. What I did was expanded (a + bi)2 into (a2 - b2) + 2abi From there, I can...
  25. N

    Understanding the Quadratic Formula and Solving Complex Equations

    Hey! In an example in my textbook, the complex equation z2 +2i*z-(1+i) = 0 is solved using the quadratic formula, so that it ends up with: z= -i ± √i = -i ± (1+i)/√2 = ...etc. Uhm.. I got 2 questions: 1) Why isn't the first solution (-i ± √i) acceptable? 2) How does √i = (1+i)/√2 ? Only...
  26. P

    Solve Complex Equation: Find Z, K in \frac{Z-a}{Z-b}=Ke^{±jθ}

    Hi, I need a little help :smile: I need to find solution for this equations: \frac{Z-a}{Z-b}=Ke^{±jθ} The Z is unknown and it is the complex number. The a and b is known and they are also complex numbers. K is the real number. I know that for -90^{°}<θ<90^{°} the graph in the complex...
  27. P

    How to get roots of this complex equation

    Homework Statement hello, i am stucked at an article from sciencedirect . somewhere it gives me the following equation and then it tells that this equation must have 4 complex roots! the variable is lambda and we want to find 4 lambda complex roots Homework Equations λ^4=0The Attempt at a...
  28. Ƒ

    Finding Solutions to a Complex Equation

    Homework Statement Solve (z+1)^5 = z^5Homework Equations None The Attempt at a Solution z^5 + 5z^4 + 10z^3 + 10z^2 + 5z + 1 = z^5 5z^4 + 10z^3 + 10z^2 + 5z + 1 = 0 5z^3(z + 2) + 5z(2z + 1) = -1 I'm not quite sure how to go about solving this. Expanding, canceling terms, and then factoring...
  29. DryRun

    Solving Complex Equation: Real & Imaginary Parts of z=x+iy

    Homework Statement Given that the real and imaginary parts of the complex number z=x+iy satisfy the equation (2-i)x-(1+3i)y=7. Find x and y. The attempt at a solution I know it's quite simple. Just equate the real and imaginary parts, but i checked and redid it again, but the answer still...
  30. F

    Finding $z$ for a Complex Equation

    How do I find all $z$ that satisfies: z = \exp\left(2+3i\right) I know the modulus has to be e^2 and the argument has to be 3 but where do I go from there?
  31. M

    MHB Can Complex Equations Be Solved More Efficiently?

    Solve ${{\left( \dfrac{1+i}{\sqrt{2}} \right)}^{x}}+{{\left( \dfrac{1-i}{\sqrt{2}} \right)}^{x}}=\sqrt{2}.$ Is there a faster way to solve this?
  32. Twinflower

    Solve Complex PI-Regulator Equation in Imaginary Plane

    Homework Statement I've set up this equation to find the integral time in a PI-regulator. \frac{\sqrt{0.02^2 - 4 \times \frac{0.02}{X}}}{2} = \frac{\pi}{100} This is not solvable by normal means because the X has to be postive, thus rendering the square root negative. I have tried by...
  33. A

    Solve for phase on the real part of a simple complex equation

    Homework Statement Solve for theta: Re { (1+j) ejθ} = -1 Homework Equations euler's formula? ejθ = cosθ + jsinθ The Attempt at a Solution (1+j) ejθ =ejθ+ Jejθ = cosθ+jsinθ +j(cosθ+jsinθ) =cosθ+jsinθ+jcosθ+j2sinθ = cosθ-sinθ+j(cosθ+sinθ) The real part equals -1 => cosθ-sinθ...
  34. M

    How Do You Solve w=0.5(z+1/z) for Complex w?

    How does one prove that for w≠+-1 , w a complex number, there are exactly 2 solutions to the equation w=0.5(z+1/z)? I'm at a total loss here. Could someone clue me in on this one?
  35. narrator

    Everything, as a complex equation

    In science, we talk about how everything is a chain of reactions. A rock passes by our planet and has its trajectory altered by the reaction with our planet. One chemical reacts with another in our brain, resulting in a chain of events. One could literally begin at the BB and follow a long...
  36. Q

    Find solutions of complex equation

    Homework Statement Give under the exponential form all non-zero solutions of (z3) + (4conjugate(z2)) = 0 Homework Equations z=x+iy where i2=-1 zn=rn(cos(nθ)+isin(nθ)) The Attempt at a Solution First i tried expanding by making z=x+iy so x3+3ix2y+4x2-3xy2-8ixy-iy3-4y2=0, but then...
  37. D

    Real Variables In Complex Equation

    Hi Homework Statement The problem asks to put the following expressions in z={ReA(e^i*)(e^i*)} Form for z=3cos(wt)-sin(wt) z=sin(wt+pi/4)+cos(wt) z=sin(wt)+2cos(wt-pi/3)-cos(wt) Homework Equations The Attempt at a Solution The problem is I don't even know which...
  38. DryRun

    Find roots of complex equation (1-x)^5 = x^5

    Homework Statement Find roots of complex equation (1-x)^5 = x^5 Homework Equations Probably Euler and/or De Moivre. The Attempt at a Solution I know i need to have z^5 on one side and all the rest on the other side. But i need 5 roots, and I'm only getting one. (1-x)^5 = x^5...
  39. B

    Solving Complex Equation arcsin(z)=w

    Hello, I am trying to solve arcsin(z)=w. Where z is a complex number z=x+iy. I got to this. p2-2*p*i*z-1=0 where p= exp(i*w) I don't know how to solve this quadratic equation. This is not homework, I got this problem out of my head. I am curious. Thank you.
  40. Q

    Is the Complex Equation z^2 - z |z| = 1 Solvable?

    Complex equation with "z" Homework Statement z^2-\bar z |z|= 1 Homework Equations Complex numbers The Attempt at a Solution(a+ib)^2-(a-ib)\sqrt{a^2+b^2} = 1 a^2-b^2+2iab-a\sqrt{a^2+b^2}+ib\sqrt{a^2+b^2} = 1 I separate the real part from the imaginary part. \left\{\begin{matrix}...
  41. Z

    Solving a Complex Equation: Finding wasym with Error

    Homework Statement i am given an equation like this wasym=sqrt[wo2+gb/2L(L-b)] where L=60.7 ± 0.4cm and b=L/5 g=9.789 ms^-2 wo=4.001 ± 0.004 −1 and i am told to find the value for wasym i get for my 4.25 rads-1 but i don't know how to calculate the error on the back of my...
  42. M

    Reshaping Complex Equations in LTI-Systems: Solving for the System Function H(z)

    Hi! New go on my problem with LTI-system which really is a math-problem: My problem i can't solve: Compute the System Function H(z) from the Frequency H(e^jw)=2*exp^(-j*3/2*w)*[cos(w/2)]^2 In other words i need to reshape the equation where i only have exp^(-j*k*w), where k is an...
  43. N

    Complex Equation Homework: Ae^(ix)=Ce^(ix) & Be^(-ix)=De^(-ix)

    Homework Statement Hi Say I have the following equation: Ae^{ix}+Be^{-ix} = Ce^{ix}+De^{-ix} then my book says that the above implies that we have the two equations Ae^{ix} = Ce^{ix} and Be^{-ix} = De^{-ix} since it must be valid for all x. I cannot see why?Niles.
  44. M

    How Do You Correctly Solve This Complex Equation?

    Homework Statement Hi, I am trying to solve the equation for I am and am having trouble Im=(Im+Ie+Ia)e^{-t/T} +Ia) Im-Ia = (Im+Ie+Ia)e^{-t/T}) ln^{Im-Ia}= Im+Ie+Ia this is were I get to but can't get the equation to work. Have I don't something incorrect in the second...
  45. L

    Can Complex Equations Be Solved Using Software?

    Homework Statement (2+5i)/(x-y)-(1-3i)/(x+y)=-7x+12i/y(square)+x(square) Homework Equations I only know the result to be x=-5/14 y=-1/14 The Attempt at a Solution If you could only lead on how to proceed I would be grateful Xbtw is there any software that could solve the above...
  46. A

    Complex equation (for a 9th grader :>)

    Warning: This is mixed with physics, though all the physics work is done. Only formatting the equation is needed. Homework Statement L is needed from this equation: L-l= 1/2 a(T-τ)2 Given: t τ l Homework Equations T=√(2L/a) a=2l/t2 The Attempt at a Solution L= 1/2 2l/t2...
  47. B

    Solving a Complex Equation with cos(wt).exp(jθ)

    Could someone help me to solve the equation below? cos(wt).exp(jθ) I want to find something like cos(wt+θ) thanks from now on
  48. D

    Solving for 'a' in a Complex Equation

    Homework Statement Solve for a 7=\frac{a}{2}(e^\frac{4}{a} + e^\frac{-4}{a}) + a Homework Equations 7=\frac{a}{2}(e^\frac{4}{a} + e^\frac{-4}{a}) + a The Attempt at a Solution 7=\frac{a}{2}(e^\frac{4}{a} + e^\frac{-4}{a}) + a multiply by 2 on both sides 14=a(e^\frac{4}{a} +...
  49. G

    Mathematica Solving a complex equation involving conjugates with Mathematica

    How can one obtain (using Mathematica) all the solutions of an equation involving conjugates like for example z \bar{z}=1. Although this equation is very simple, it seems that Mathematica is not able to handle it. I obtain: Solve[z Conjugate[z] == 1, z] InverseFunction::ifun: Inverse...
  50. icystrike

    Solving a Complex Equation: Express as e^(iθ)

    Homework Statement Express the complex number in exp. form -\frac{1}{2}(1+i\sqrt{3}) Solve the following eqn: (w+2)^{4}=-\frac{1}{2}(1+i\sqrt{3}) Homework Equations The Attempt at a Solution e^{i\frac{\pi}{3}} w+2=e^{(\frac{\frac{\pi}{3}+2k\pi}{4})i}...
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