What is Complex plane: Definition and 128 Discussions

In mathematics, the complex plane or z-plane is the plane associated with complex coordinate system, formed or established by the real axis and the perpendicular imaginary axis. It can be thought of as a modified Cartesian plane, with the real part of a complex number represented by a displacement along the x-axis, and the imaginary part by a displacement along the y-axis.The concept of the complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers can be expressed most easily in polar coordinates—the magnitude or modulus of the product is the product of the two absolute values, or moduli, and the angle or argument of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation.
The complex plane is sometimes known as the Argand plane or Gauss plane.

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  1. B

    Question on graphing in the complex plane

    Okay, I need to graph the following set in the complex plane: M={z\inC:[(1<|z-i|\leq2) and (z\neq2+i)] or [z = 1 + \pii]} I got the last two constraints, but the first one is what's giving me trouble. is z-i just x+yi that is (1,1) on the complex plane lowered by 1? Thanks
  2. N

    How Do Complex Plane Transformations Affect Equations?

    http://img34.imageshack.us/img34/5391/13262160.jpg http://g.imageshack.us/img34/13262160.jpg/1/ http://img46.imageshack.us/img46/7397/62501858.jpg http://g.imageshack.us/img46/62501858.jpg/1/ http://img7.imageshack.us/img7/2651/15142727.jpg...
  3. B

    Isolated singularity in complex plane

    Homework Statement Locate each of the isolated singularities and tell whether it is a removable singularity, a pole, or an essential singularity. If removable, give the value of the function at the point. If a pole, give the order of the pole. f(z) = \pi Cot(z\pi) Homework Equations...
  4. I

    Conditions on complex plane wave solutions to Maxwell's Equations

    Homework Statement What conditions need to be imposed on \vec{E}0, \vec{B}0, \vec{k} and ω to ensure the following equations solve Maxwell's equations in a region with permittivity ε and permeability µ, where the charge density and the current density vanish: \vec{E} = Re{ \vec{E}0...
  5. C

    How Do You Perform a 90 Degree Rotation of a Complex Number Around a Point?

    I'm reading this book on modern geometry and I was wondering if I'm doing these problems right: if I'm give a point 2+i and I'm suppose to rotate is 90 degrees first I move it to the origin T(z)=z-(2+i) second, I rotate it e^(pi/2*i)*z I'm not sure how to interpret that...
  6. H

    Understanding Branch Cuts for Analytic Functions in the Complex Plane

    I have a question with regards to branch cuts: Say I have a function f(z) = log(z2 - 1). Why is a simple branch cut connecting z = -1 and z = +1 not sufficient to define an analytic function? On the other hand, why is it sufficient for the function f(z) = (z^2 - 1)^(1/2) ? This is in the...
  7. G

    Conformal mappings on the complex plane

    Homework Statement a) Using the conformal mapping w=cosh(z), find a rectangle R in the z-plane which maps to the region in the w-plane with boundaries as follows: - a plate of constant temperature on the line segment {w=u+iv : -1<u<1, v=0} - an outer boundary of cooler constant temperature...
  8. B

    Complex plane graphing program?

    anyone no of a free downloadable program the graphs the complex plane. its just I am not to bright but i have to do this assingment on conformal mapping and i can't rearange these equations correctly into rcisthetre form, i like doing things as exact as i can make them and if i did it by hand i...
  9. C

    Understanding the Complex Plane and Real Plane Relationship

    I may be asking a stupid question, but what is the co-relation between the complex plane and the real plane? I know Euler's equation ei\pi+1=0 relates them, but graphically, how are they related?
  10. E

    Find a+b in Regular Hexagon Complex Plane Problem

    Homework Statement A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let R be the region outside the hexagon, and let S = \{ 1/z |x \in R} . Then the area of S has the form a \pi...
  11. E

    Topological Properties of Closed Sets in the Complex Plane

    [SOLVED] Topological Properties of Closed Sets in the Complex Plane Homework Statement 1. Show that the boundary of any set D is itself a closed set. 2. Show that if D is a set and E is a closed set containing D, then E must contain the boundary of D. 3. Let C be a bounded closed convex set...
  12. E

    Mathematica Mathematica plot of a function from [0,1] to the complex plane

    If I have a function from [0,1] to the complex plane, is there a way that I can plot it with Mathematica 6.0?
  13. S

    What is the proof of the theorem on limit points in topology?

    I'm having some trouble understanding the distinction between closed sets, open sets, and those which are neither when the set itself involves there not being a finite boundary. For example, the set { |z - 4| >= |z| : z is complex}. This turns out to be the inequality 2>= Re(z). On the right...
  14. Simfish

    How do singularities of a function on a complex plane affect real line behavior?

    consider the function \frac{1}{\epsilon^2 + z^2} So we know that there are two poles, one at z = i \epsilon, one at z = - i \epsilon. So when this function never hits 0 on the real line, how do the singularities affect its behavior on the line? Okay, so poles are a subclass of singularities...
  15. K

    Could a Non-Euclidean Complex Plane Be Defined Using Differential Geometry?

    do not know if such generalization exist, my question is... we have Euler identity e^{ix}=cos(x)+isin(x) considering that complex plane defined by real and complex part, is Euclidean (a,b) but could we define using the axioms and tools of Differential Geommetry a Non-Euclidean complex...
  16. R

    Radius on convergence in the complex plane

    Anyone knows anything about the Radius on convergence in the complex plane (Complex Analysis)
  17. M

    Surjectivity of Sin/Cos over the Complex Plane

    Verify Identities Cos(z) = (exp(iz)+exp(-iz))/2 sin(z)= (exp(iz)-exp(-iz))/2i Using identities show that sin(z) and cos(z) are surjective from C to C. Determine all z in C such that sin(z) = 12i/5 Solutions: The verification seems simple using eulers formula and the even/odd nature of...
  18. M

    Boundedness in extended complex plane

    what does it mean to be bounded in extended complex plane (say C')? can we say all subsets of C' is bdd because C' itself bdd?
  19. S

    Help with Describing Image of |z-1| ≤ 1 in Complex Plane

    Hi everyone - I'm sure there's somebody here who can help with such a trivial question. It's not a homework question before that's assumed - it's from a past exam paper, which I'm using for revision, sans answers. It asks to describe, in the complex plane, the image of: |z - 1| \leq 1...
  20. E

    Images under exp in complex plane

    Describe the image under exp of the line with equation y = x. To do this you should find an equation (at least parametrically) for the image (you can start with the parametric form x = t; y = t), plot it reasonably carefully, and explain what happens in the limits as t approaches infinity and t...
  21. T

    Complex Plane Dimension Number

    What dimension between space-time and 11 Dimensions is allocated to the complex plane 'visualized' and used in complex number theory? Complex numbers are used in every branch of maths and physics, based on an imaginary complex plane, z = x +iy where y is an imaginary axis and i^2=-1. - It's...
  22. Y

    Trig chart for the complex plane

    i was trying to make a trig chart for the complex plane and accindently found 0>1. at pi/4, the hypotenuse is 0, but the real leg is 1. how is that? imaginary trig must be fun. at \theta=\frac{\pi}{4}, i got: sin=undefined cos=undefined tan=i csc=0 sec=0 cot=-i it is in the first...
  23. B

    Locus of Points in Complex Plane for Constant a: Q

    Q. For the real constant a find the loci of all points z = x + yi in the complex plane that satisfy: a) {\mathop{\rm Re}\nolimits} \left\{ {\log \left( {\frac{{z - ia}}{{z + ia}}} \right)} \right\} = c,c > 0 b) {\mathop{\rm Im}\nolimits} \left\{ {\log \left( {\frac{{z - ia}}{{z + ia}}}...
  24. cepheid

    A Question About the Complex Plane

    I'm taking an introductory course in Complex Analysis. Close to the beginning of the term, we had a review of complex numbers, but we weren't expected to know any of this crazy number theory stuff, and as far as the reasons for this addition to the real numbers, my prof offered only this...
  25. E

    Is there a numerical method to solve complex plane integrals?

    Let be the integral in the complex plane: Int(1-i8,1+i8)dz/f(z) is there anumerical method to solve it (i know that it could be solved by calculating the residues of f(z) buit my iterest is in knowing if it can be calculated by using some numerical method...thanks)
  26. S

    ElementaryFunctions/Exp/06/GraphingOnTheComplexPlane

    During a conversation I had yesterday, a math professor I occasionally meet with mentioned in passing, "and you might want to try to graph f(z)=e^z on the complex plane...hm...yes...anyway..." (where "z" is complex). So I sat down at Taco Bell yesterday to think about it, and, for a few minutes...
  27. S

    What is the radius of a circle represented by the equation |z - z1|=k|z - z2|?

    In general, |z - zo|=r, where z_o is a fixed point and r is a positive number, represents a circle centered at z_o and with radius r. |z - z1|=k|z - z2|, where z_1 and z_2 are fixed points, also apparently represents a circle, except maybe in the case where k=1. Then we have a line, or a...
  28. pellman

    Dirac delta function on the complex plane?

    Supposedly, &int; ez*(z - z0)f(z) dz*dz is proportional to f(z0) much in the same way that (1/2&pi;)&int; eiy(x - x0)f(x) dxdy = &int; &delta;(x - x0)f(x) dx = f(x0) Is this true? Could someone help convince me of it, or point me to a text? I would say that even if true, it...
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