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.

View More On Wikipedia.org
Recent contents Top users
  1. A

    Real integral in complex plane

    Homework Statement I want to find the value of the integral: ∫cos(x)/((x+a)2+1) dx from ]-∞;∞[ Homework Equations Residue theorem The Attempt at a Solution My question is seeking more a conceptual understanding of why transforming to the complex plane works. According to...
  2. M

    Conic Sections on the Complex Plane (circle)

    Homework Statement Describe the locus and determine the Cartesian Equation of: \left|z-3-5i\right|= 2 Homework Equations \left|z-C\right|= r -----> formula for a circle on complex plane Where C = the centre z = the moving point (locus) (x-h)^{2}+(y-k)^{2}=r^{2} -----> Formula...
  3. A

    Regions of the complex plane - finding the locus

    Find the locus defined by |z-2|-|z+2|=3 The given example rewrites the left hand side as: \sqrt{(x-2)^2+y^2}-\sqrt{(x+2)^2+y^2} 1) When they rewrite it as that, they square it and square root it right? Why isn't it squaring the whole expression and rooting it (isn't that the rule?)...
  4. C

    Plotting the roots of unity on the complex plane

    Homework Statement Find the 6th complex roots of √3 + i. Homework Equations z^6=2(cos(π/6)+isin(π/6)) r^6=2, r=2^1/6 6θ=π/6+2kπ, θ=π/36+kπ/3 The Attempt at a Solution When k=0, z = 2^1/6(cos(π/36)+isin(π/36)), When k=1, z = 2^1/6(cos(13π/36)+isin(13π/36)), When k=2, z =...
  5. B

    Integration in complex plane - deforming contours

    Homework Statement The following sum of integrals has integrals that are both integrated over straight lines in the complex plane. Deform the contours back to the origin and avoid the singularity at x = infinity to prove the integral formula, \int_{ - {x_0}}^\infty {(x + {x_0}){e^{ -...
  6. M

    Vector field curvature in the complex plane

    Hey all, I have a vector field described by a complex potential function (so I have potential lines and streamlines). I am looking for a way to express its curvature at every point, but I can't find such a formula in my books. I have searched in wikipedia and I read that the way to define it...
  7. J

    MHB Locus in the complex plane.

    Area of Region Bounded by the locus of $z$ which satisfy the equation \displaystyle \arg \left(\frac{z+5i}{z-5i}\right) = \pm \frac{\pi}{4} is
  8. G

    Find the integer that is nearest to the area of complex plane A

    Consider the region A in the complex plane that consists of all points z such that both \frac{z}{40} and \frac{40}{\overline{z}} have real and imaginary parts between 0 and 1, inclusive. What is the integer that is nearest the area of A? Let z = a + bi and \overline{z} = a - bi a = real part...
  9. R

    Complex analysis: Sketch the region in the complex plane

    Homework Statement Sketch: {z: \pi?4 < Arg z ≤ \pi} Homework Equations The Attempt at a Solution Is it right to assume z0 = 0 ; a = a (radius = a) ; and taking \alpha = \pi/4 ; \beta = \pi And now in order to sketch the problem after setting up the complex plane is it correct...
  10. D

    MHB Drawing a Clockwise Rectangle on the Complex Plane with Tikz

    How can I draw a rectangle oriented clockwise on the complex plane with vertices on (0,0), (0,4), (10,4), and (10,0)? I am guessing the tikz package needs to be used but I am not skilled in making pictures.
  11. N

    Complex Analysis - Sketching regions in a complex plane

    Homework Statement |2z -1|\geq|z + i| The Attempt at a Solution The problem I have with this one is the 2z, I just need a clue on how to go about centering this one. If it were just |z - 1|; z_{0} would be 1.
  12. P

    (Algebra) Isometries on the complex plane

    So this is the problem as written and I'm totally lost. Any help or explanation would be greatly appreciated. "Viewing ℂ=ℝ2 , we can identify the complex numbers z = a+bi and w=c+di with the vectors (a,b) and (c,d) in R2 , respectively. Then we can form their dot product...
  13. P

    How Do I Sketch Complex Regions in the Complex Plane?

    Homework Statement I do not have specific problem, I am struggling in my complex variables class and I think a large part of it is because I struggle at sketching regions in ℂ. For instance let z=x+ I full understand what |z|< 1 looks like and all that (punctured disk, things in that...
  14. G

    Euclidean geometry and complex plane

    Can someone please describe to me how Euclidean Geometry is connected to the complex plane? Angles preservations, distance, Mobius Transformations, isometries, anything would be nice. Also, how can hyperbolic geometry be described with complex numbers?
  15. F

    Motion of Point P in Complex Plane: Finding z(t)

    The motion of a point P in the complex plane is defined by the principal root of z^5= (1+ t)^i a)find z(t) b)Show that P is undergoing a circular motion. Find the velocity and acceleration as a function of time I'm pretty sure I know how to do b but I don't really understand the...
  16. P

    Sequences in Complex Plane which Converge Absolutely

    Let A be a non-empty subset of the complex plane and let b ∈ ℂ be an arbitrary point not in A. Now define d(A,b) := inf{|z-b| : z ∈ A}. Show that if A is closed, then there is an a ∈ A such that d(A,b) = |a-b|. Ok so basically what I did was begin by choosing some arbitrary element of A and...
  17. 1

    Is the complex plane meaningful?

    I'm not sure I understand the complex plane very well. For the cartesian plane, or other planes such as polar, points are plotted by a function. One value of x coresponds to a value of y. (or r to theta, or whatever.) The complex plane isn't a plot of functions, just of a single number...
  18. B

    Use suitable contours in the complex plane and the residue theorem to show that

    Homework Statement Use suitable contours in the complex plane and the residue theorem to show that integral from -infinity to +infinity of [1/(1+(x^4))] dx=pi/(sqrt(2)) Fix R > 1, and consider the counterclockwise-oriented contour C consisting of the upper half circle of radius R...
  19. Telemachus

    Two different demonstrations on the complex plane

    Hi there, I have to prove this two sentences . I think I've solved the first, but I'm quiet stuck with the second. The first says: 1) Demonstrate that the equation of a line or a circumference in the complex plane can be written this way: \alpha z . \bar{z}+\beta z+\bar{\beta}...
  20. W

    Simple-Connectedness in Complex Plane: Def. in Terms of Riemann Sphere.

    Hello, There is a definition of simple-connectedness for a region R of the complex plane C that states that a region R is simply-connected in C if the complement of the region in the Riemann Sphere is connected. I don't know if I'm missing something; I guess we are actually consider...
  21. B

    Finding Omega: Evaluating sin^(-1)(3) on the Complex Plane

    Homework Statement if z= sin (omega) find an expression for omega as a function of z that can be used to evaluate all possible values of sin^(-1) (3). Plot these values on the complex plane The Attempt at a Solution z= sin (omega) 3= sin (omega) I don't know how to proceed from...
  22. J

    Exploring the Geometric Properties of Complex Ratios

    I am reading Visual Complex Analysis by Dr. Tristan Needham and am hung up on some of the geometrical concepts. In particular, I am having trouble with ideas involving the geometric properties of numbers like: \frac{z-a}{z-b} Note: I am still in the first and second chapters, which deal...
  23. B

    Sketch on the complex plane the region where the following two power series both

    Homework Statement sketch on the complex plane the region where the following two power series both converge 1) sigma from n=0 to infinity [(z-1)^n]/[n^2] 2) sigma from n=0 to infinity [((n!)^2)((z+4i)^n)]/[2n]! The Attempt at a Solution R=lim as n tends to infinity...
  24. T

    Complex analysis and complex plane

    Homework Statement Let z= x + yi be a complex number. and f(z) = u + vi a complex function. As: u = sinx\astcoshy v= cosx\astsinhy And if z has a trajectory shown in the attached image. What would be the trajectory of the point (u,v) ?
  25. F

    Another way to extend the Complex Plane (Insertsomethingthatgetsyourattention)

    Hey guys so I was thinking about how to extend the Complex Plane out to a third dimension and I started reading the whole tidbit about Quaternions and their mechanics when I realized that I want to propose a whole new question. Now please feel free to prove me wrong if you can answer it because...
  26. M

    Showing a function is bounded in the complex plane.

    Homework Statement Hi everyone. I must show that if f is a continuous function over the complex plane, with limit as z tends to infinity = 0, then f is in fact bounded. The Attempt at a Solution Since f is continuous and lim z --> infinity f(z) = 0, by definition of limit at infinity I know...
  27. E

    Fourier Transform and Complex Plane

    I have been playing with the FFT and graphs. The easiest example I could think of for a transform was the top hat function (ie 0,0,0,0,0...1,1,1...0,0,0,0,0). When I transform this from the time domain to the frequency domain, it returns a sinc function when I take the absolute value squared of...
  28. M

    Graph inequality in complex plane; negative z value

    Homework Statement Graph the following inequality in the complex plane: |1 - z| < 1 2. The attempt at a solution In order to graph the inequality I need to get the left side in the form |z - ...| |1 - z| < 1 |(-1)z + 1| < 1 |-1(z - 1)| < 1 |-1||z - 1| < 1 (1)|z - 1| < 1 |z - 1| < 1...
  29. P

    Does an antiderivative of e^z/z^3 exist in the punctured complex plane?

    Hi, so my question is the subject line. In the multiply connected domain |z|>0, does the function f(z) = e^z/z^3 have an antiderivative? I'm learning from Brown and Churchill, and they have a theroem on pg. 142 that leads me to believe it does. I don't remember what my prof said about this...
  30. M

    Visualizing Complex Numbers on the Complex Plane

    Homework Statement Problem 1. Create a program to display a complex number (or a list of complex numbers) as an arrow (or arrows) on the complex plane. i know RandomComplex[] will give me a random complex number, and i know RandomComplex[{1 - I, 1 + I}, 5, WorkingPrecision ->...
  31. R

    How do you parametrize the unit square in the complex plane?

    My book just gives me what each individual piece is but doesn't explain anything.
  32. L

    Proving f(z) is a continuous function in the entire complex plane

    Homework Statement Show that the function f(z) = Re(z) + Im(z) is continuous in the entire complex plane. Homework Equations The Attempt at a Solution I know that to prove f(z) is a continuous function i have to show that it is continuous at each part of its domain. I take...
  33. W

    HELP Absolute Values on a Complex Plane

    Homework Statement Draw |z| on a complex plane, where z = -3+4i Homework Equations N/A The Attempt at a Solution [PLAIN]http://img530.imageshack.us/img530/1786/aaakr.jpg Can anyone please tell me which answer is correct? Both of them have a moduli of 5. So should the circle...
  34. T

    [Complex plane] arg[(z+i)/(z-1)] = pi/2

    Homework Statement Sketch the set of complex numbers z for which the following is true: arg[(z+i)/(z-1)] = \pi/2 Homework Equations if z=a+bi then arg(z) = arctan(b/a) [1] and if Z and W are complex numbers then arg(Z/W) = arg(Z) - arg(W) [2] The Attempt at a Solution using eq. [2] i...
  35. Somefantastik

    Finite Set of Points in Complex Plane: $\{e^{n r \pi i}\}$

    Homework Statement \left\{ e^{n r \pi i}: n \in \textbf{Z} \right\} , r \in \textbf{Q} I'm trying to show that this set is finite. Homework Equations The Attempt at a Solution Other than the fact that these points lie on the unit circle in the complex plane, I'm not sure...
  36. A

    Definition of compactness in the EXTENDED complex plane?

    Definition of "compactness" in the EXTENDED complex plane? How does one define a compact set in the extended complex plane \mathbb C^* = \mathbb C \cup \{ \infty \}? "Closed and bounded" doesn't really make sense anymore, as I'm assuming it's permissible for a compact set to contain the point...
  37. S

    Understand the definition of a circle in the complex plane

    Homework Statement I know following that |z| = 1 where z \in \mathbb{C} is the definition of unit circle in the complex plane. then if the exist another complex number c which lies within the distance r from z then distance from the two numbers kan be discribe as |z-c| = r If...
  38. Z

    Can a finite polynomial have no roots on the left of the complex plane?

    given a finite polynomial a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+...+a_{n}x^{n} =P(x) is there a theorem or similar to ensure that P(x) has NO roots on the left of complex plane defined by Re(x<0) ??
  39. W

    Continuation of a real function into the total complex plane

    suppose i have a real function f=f(x) this function is smooth everywhere on the real line for example, f=e^x. The problem is, is the continuation of the function into the complex plane unique? if so, does it hold that f(z)=f(z*)*?
  40. Z

    Equation of a vertical line in the complex plane

    Homework Statement I need to solve \int_L \bar{z}-1 where L is the line from 1 to 1+2i. Homework Equations The Attempt at a Solution I know that I need to set z equal to the equation of the line and then integrate, but in this case I'm not sure how to express the equation of...
  41. M

    Branch Cuts in the Complex Plane

    Homework Statement The function f(z) = (1-z2)1/2 of the complex variable z is defined to be real and positive on the real axis in the range -1 < x < 1. Using cuts running along the real axis for 1 < x < infinity and -infinity < x < -1, show how f(z) is made single-valued and evaluate...
  42. M

    Graph Curves in the Complex Plane

    Homework Statement [/b] Graph the locus represented by the following. \left|z+2i\right| + \left|z-2i\right| = 6 Homework Equations The Attempt at a Solution z = x + iy so z-2i = x + (y-2)i and z+2i = x + (y-2)i So I have: sqrt(x^2 + (y-2)^2) + sqrt(x^2 + (y+2)^2) = 6...
  43. N

    Integration RUNGE_KUTTA in complex plane.

    Hi, I have one serious problem while solving rayleigh equation using blasius profile, in which so as to remove the singularity the intergration contour is defined in a complex plane. 4th order runge kutta is used but if the step size( h) is is in complex, it is giving some error. would anyone...
  44. L

    What is the significance of symmetry in the complex plane?

    How does one express mathematically the fact that: if we complex-conjugated everything (switch i to -i (j to -j etc. in hypercomplex numbers) in all the definitions, theorems, functions, variables, exercises, jokes ;-)) in the mathematical literature the statements would still be true?
  45. J

    Orders of poles in the complex plane.

    Homework Statement For a function f(z)= [e^(2*pi*i*a*z)] / [1 + z^2] I need to find the order of the poles at i and -i. (I'm pretty sure these are the only poles.) Homework Equations The Attempt at a Solution I'm not totally clear on how I go about finding the orders. I have a...
  46. Z

    Zeros of functions on the complex plane

    what is the relationship (if any) of the following statement - A function has ALL the zeros on the line (complex plane) Re (z) = A for some Real A - A function has ALL the zeros on the unit circle defined by |z| \le 1 i think there is a transformation of coordinates so the line Re...
  47. J

    Complex analysis - graphing in complex plane

    Homework Statement Graph the following in the complex plane {zϵC: (6+i)z + (6-i)zbar + 5 = 0} Homework Equations z=x+iy zbar=x-iy The Attempt at a Solution Substituting the equations gives 2(6x-y) + 5 = 0 => y = 6x + (5/2) But that's a line in R^2. The imaginary parts...
  48. A

    I don't get branch cuts in the complex plane at ALL

    Suppose you're trying to provide a branch cut in \mathbb{C} that will define a single-value branch of f(z) = \log(z - z_0). I don't know where to begin. Can someone help explain this concept to me?
  49. B

    Why can exp function in complex plane be defind as e^x(cosy+i siny)

    Hi, I know this one is stupid, but i am still confused. why e^(iy) = cosy + i siny? thank you.
  50. C

    Identify and sketch the region in the complex plane satisfying

    Homework Statement Identify and sketch the region in the complex plane satisfying | \frac{2 z - 1}{z + i} | \geq 1 Homework Equations The Attempt at a Solution
Back
Top