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. S

    I Curve of zeta(0.5 + i t) : "Dense" on complex plane?

    This is a discussion on MathOverflow where a conjecture is discussed that the curve of ##\zeta(0.5+it)## is "dense" on the complex plane. https://mathoverflow.net/questions/73098/negative-values-of-riemann-zeta-function-on-the-critical-line From a couple of sources, e.g...
  2. R

    A straight line in the complex plane

    sz+tz*+r=0=say w so w* = s*z* + t*z + r*=0 Now , w+w* = (s+t*)z + (t+s*)z* + r+r* = 0 = p*z + pz* + k = 0...eq(1) ( k is a constant or twice real part of w) which is in complex straight line equation form i.e ab* + a*b + c = 0 ( a,b are complex number and c a real number. Now, again...
  3. A

    I Integration being unchanged after rotation

    This question is about the general 1 loop correction to the propagator in QFT (this is actually not important for this question). Let's say we have an integral over an integration variable x, and this x ranges from ##-\infty## to ##\infty##. If we look at this integration contour in the complex...
  4. A

    Online app which plots F(z) in the complex plane

    I am looking for an app that can instantaneously plot the function f(z) in the complex plane once z is given. It would be much favorable if this process is fast which allows one to visualize f(z) when the user is moving the mouse on the complex plane to the location of z. One possible...
  5. C

    MHB Set of points on the complex plane

    Dear Everybody, I am wanting to check the solution to this question: Sketch the set of points determined by the given conditions: a.) $\left| z-1+i \right|=1$ b.)$\left| z+i \right|\le3$ c.)$\left| z-4i \right|\ge4$ work: I know (a.) is a circle with radius 1 and its center at (-1,1) on the...
  6. Felipe Lincoln

    Proving that sin(z1+z2)=sinz1cosz2+sinz2cosz1 in complex plane (Arfken)

    Homework Statement Prove that ## \sin(z_1+z_2) = \sin z_1\cos z_2+\sin z_2\cos z_1## such that ##z_1,z_2\in\mathbb{C}## Homework Equations ##\sin z = \sum\limits_{n=1, \mathrm{ odd}}^\infty (-1)^{(n-1)/2}\dfrac{z^n}{n!} = \sum\limits_{s=0}^\infty (-1)^s\dfrac{z^{2s+1}}{(2s+1)!}## ##\cos z =...
  7. Tspirit

    Fourier transform in the complex plane

    Homework Statement I am reading the book of Gerry and Knight "Introductory Quantum Optics" (2004). In page 60, Chapter 3.7, there is two equation referring Fourier Transformation in the complex plane as follows: $$g(u)=\int f(\alpha)e^{\alpha^{*}u-\alpha u^{*}}d^{2}\alpha, (3.94a)$$...
  8. S

    Understanding Complex Plane and Finding Arguments: A Scientist's Perspective

    Homework Statement The picture below. Homework Equations cos2x=1-2sinx sin2x= 2sinxcosxThe Attempt at a Solution I got the modulus by using the Pythagoras theorem which is 2sin theta But I faced difficulty to find the argument. I have no idea why i end up with tan a (alpha) = cot theta which...
  9. A

    MHB Mapping of a Circle in the Complex Plane

    I have a circle with centre (-4,0) and radius 1. I need to draw the image of this object under the following mappings: a) w=e^(ipi)z b) w = 2z c) w = 2e^(ipi)z d) w = z + 2 + 2i I have managed to complete the question for a square and a rectangle as the points are easy to map as they are...
  10. Math Amateur

    MHB Theorem 1.8: Sets or Domains in the Complex Plane - Palka Ch.2

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter 2: The Rudiments of Plane Topology ... I need help with an aspect of Theorem 1.8 ... Theorem 1.8 (preceded by its "proof") reads as follows...
  11. W

    Complex Plane Homework: Mobius Transformation Advice

    Homework Statement Homework EquationsThe Attempt at a Solution I'm not sure how to even begin this problem. My notes mentioned something about a Mobius Transformation but that's not something that I've been taught, and certainly not something I'm familiar with. Any advice would be greatly...
  12. S

    I Identical zero function in the complex plane

    Hi! If a holomorphic function ##f:G \to C##, where ##G## is a region in the complex plane is equal to zero for all values ##z## in a disk ##D_{[z_0,r]}##, inside ##G##, is it zero everywhere in the region G? And if this is true, does it mean that if an entire function is zero in a disk, it is...
  13. C

    Derivative in the complex plane

    Homework Statement f(z)=2x^3+3iy^2 then it wants f '(x+ix^2) The Attempt at a Solution So I take the partial with respect to x and i get 6x^2 then partial with respect to y and I get 6iy, then I plug in x for the real part and x-squared for the imaginary part, then I get f '...
  14. Poetria

    Complex functions with a real variable (graphs)

    Homework Statement How do the values of the following functions move in the complex plane when t (a positive real number) goes to positive infinity? y=t^2 y=1+i*t^2[/B] y=(2+3*i)/t The Attempt at a Solution I thought: y=t^2 - along a part of a line that does not pass through the...
  15. C

    Does the Limit in the Complex Plane Approach Infinity?

    Homework Statement lim as z--> i , \frac{z^2-1}{z^2+1} The Attempt at a Solution [/B]When we plug in i we get -2/0, so we get division by 0, Does this mean the limit is infinity, I also tried approaching from z=x+i where x went to 0, you get the same answer, I also approached from...
  16. C

    Mapping a Circle in the Complex Plane using f(z)=1/z

    Homework Statement What is the mapping of the circle of radius 1 centered at z=-2i under the mappinf f(z)=1/z The Attempt at a Solution I write the circle in polar form -2i+e^{ix} Now we invert it and multiply by the complex conjugate. so we get f(z)=...
  17. C

    Complex Conjugates in Quadratic Equations: Solving for z

    Homework Statement Solve each equation for z=a+ib z^{*2}=4z where z* is the complex conjugate The Attempt at a Solution I wrote z and z* in terms of x and iy , and tried solving for x and y, but I get quartic terms for y, it doesn't look like it will boil down, It was like over 2 pages of...
  18. MickeyBlue

    I Sketching Complex Numbers in the Complex Plane

    I've just had my first batch of lectures on complex numbers (a very new idea to me). Algebraic operations and the idea behind conjugates are straightforward enough, as these seem to boil down to vectors. My problem is sketching. I have trouble defining the real and imaginary parts, and I don't...
  19. Destroxia

    Why are complex numbers represented on a plane?

    So I know that a complex number can be represented by ##z=x+iy##, where ## z = x + iy \in \mathbb{C}##. Would it be okay to then state that ## z = x + iy \in \mathbb{C} := (x,y) \in \mathbb{R}^2 ##? If we can just look at complex numbers as coordinates in ##\mathbb{R}^2## what is the point of...
  20. Z

    Complex Plane - Graphing Powers

    Homework Statement If W is represented as the point shown in blue which of the other points satisfy z=Sqrt[w]? Homework Equations The Attempt at a Solution (The answer is Z2)[/B] I'm trying to study for a test and this is a practice problem and the book doesn't go into great detail about...
  21. 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...
  22. davidbenari

    ##\int \frac{dz}{z} ## along a line on the complex plane

    Some time ago I stumbled upon the integration ##\int \frac{dz}{z} ## along a line on the complex plane. I was confused because ##Ln(z)## is a multivalued function but apparently the way you do it is by only considering the principal branch from ##[-\pi,\pi]##. But I don't understand this at...
  23. K

    MHB Image of the upper half complex plane, under the function g(z) = exp(2 \pi i z).

    Problem: Given $W = \{z: z=x+iy, \ y>0\}$ and $g(z) = e^{2 \pi i z},$ what does the set $g(W)$ look like, and is it simply connected? Attempt: $W$ represents the upper-half complex plane. And $$g(z) = e^{2 \pi i (x+iy)} = \cdots = e^{-2\pi y}(\cos (2 \pi x) + i \sin (2 \pi x)).$$ (Am I on the...
  24. M

    Set of Points in complex plane

    Homework Statement Describe the set of points determined by the given condition in the complex plane: |z - 1 + i| = 1 Homework Equations |z| = sqrt(x2 + y2) z = x + iy The Attempt at a Solution Tried to put absolute values on every thing by the Triangle inequality |z| - |1| + |i| = |1|...
  25. dwn

    Navigating a Complex Plane Curve: A Homework Guide

    Homework Statement Attached Image Homework Equations this is not a simple plane curve or a close plane curve so I use the formula: ∫ F ⋅ dr/dt dt The Attempt at a Solution From the point (0,0) to (2,4) Direction Vector v(t) = <2-0, 4-0> Parametric Equation: r(t) = (2t + 0) i + (4t + 0) j...
  26. B

    Continuity of an arc in the complex plane

    Hello everyone, I have a rather simple question. I have the curve ## C(t) = \begin{cases} 1 + it & \text{if}~ 0 \le t \le 2 \\ (t-1) + 2i & \text{if }~ 2 \le t \le 3 \end{cases} ## which is obviously formed from the two curves. This curve is regarded as an arc if the functions ##x(t)## and...
  27. PcumP_Ravenclaw

    Doubt about rotation isometry on the complex plane

    Dear All, In the 2nd paragraph of the attachment can you please explain to me why we are trying to make ## r(z) = z ## and what does "As r is not the identity..." mean?? and how did the line ## L = 0.5*b + ρe^{θ/2} ## come about? Danke...
  28. PcumP_Ravenclaw

    Derivation of rotation isometry on the complex plane

    Dear all, can you please verify if my derivation of the algebraic formula for the rotation isometry is correct. The handwritten file is attached. The derivation from the book (Alan F beardon, Algebra and Geometry) which is succinct but rather unclear is given below. Assume that f (z) = az + b...
  29. DiracPool

    Rotations in the complex plane

    I'm trying to check my understanding of rotations in the complex plane. Do I have any of this wrong? If so, can you please explain why? 1) Rotations a) Say we start with a vector, Q, defined on the real number line as (5,0). If I multiply that vector by i, we now have a vector "iQ" that...
  30. M

    The Geometry of Non-Euclidean Complex Planes

    If the Parallel Axiom is just one of several possible assumptions, why is it that so many mathematical relationships seem to only be expressible in the Euclidean plane? Do planes with positive or negative curvature give analogues to the Agrand plane for complex algebra, or the Cartesian plane...
  31. N

    Connectedness of a Given Set in the Complex Plane

    Homework Statement Let A=\mathbf{C}-{z:Re(z) and Im(z) are rational}. Show that A is a connected set. Homework Equations My book gives the definition of a disconnected set as a set that satisfies three conditions. A set A is disconnected if there exist two open sets U and V in \mathbf{C}...
  32. T

    Roots of the Following Series in the Complex Plane

    Homework Statement For the series x^n - x^(n-1) - x^(n-2) ... - x^(0) the roots seem to be x = 2 and the circle around the complex plane with radius i or 1 I'm not sure how you would say it as n approaches infinity. Here's an image of the roots where n = 15...
  33. T

    [Complex plane] arg[(z-1)/(z+1)] = pi/3

    Homework Statement Show that arg[(z-1)/(z+1)] represents a circle. Find it's radius and centre. Homework Equations The Attempt at a Solution using z = (x+iy) i narrowed down to (z-1)/(z+1) = (iy)/(1+x) , assuming it was a circle. What next? Is this correct approach??
  34. applestrudle

    Complex plane locus question (another one)

    Homework Statement again there is no answer provided in the book! a +bt +ct^2 = z where t is a real parameter, and a, b, c are complex numbers with b/c real Homework Equations The Attempt at a Solution b/c real indicates that b and c are pure imaginary so when you split the...
  35. applestrudle

    Find the locus in the complex plane of points that satisfy

    Homework Statement find the locus in the complex plane that satisfies z -c = p (1+it/1-it) c is complex, p is real t is a real parameter Homework Equations The Attempt at a Solution there is no answer in the textbook so i wanted to check my answer. I got a unit circle...
  36. J

    What is the concept of the extended complex plane/Riemann Sphere?

    In a report I am writing I want to define the extended complex plane/Riemann Sphere and I would like to check if I grasp the concept properly: Consider the Euclidean space \mathbb{R}^3 where the x-y plane represents \mathbb{C}. Consider the sphere with south pole (0,0,0) and north pole...
  37. G

    Understanding Complex Plane Regions: Solving Equations and Graphing Circles

    Homework Statement Shade each region in the complex plane. Justify your solution. 1) z - Conjugate[z] = 4 2) 1 + z, where |z| = 1 The Attempt at a Solution So for my attempt for 1 is: Let z = x + iy therefore Conjugate[z] = x - iy z - Conjugate[z] = 4 x + iy - (x - iy) = 4...
  38. S

    Sketch the region of the complex plane

    Homework Statement Sketch the region of the complex plane specified by: |z - 4 + 3i| ≤ 5 Homework Equations The Attempt at a Solution I have tried re-writing the modulus as √[(z)^2 (- 4)^2 + (3i)^2] and from this I have managed to arrive at z ≤ 3√2 But not sure if I needed...
  39. S

    MHB Equation of a Circle in the Complex Plane

    Hi Guys can you please help me out for the following question: Show that the equation of the circle $$\gamma(a;r)$$ centered at $$a\in\mathbb{C}$$ and radius $$r$$ can be written in the form: $$|z|^2 - 2Re(\bar{a}z) + |a|^2 = r^2 $$
  40. N

    Derivative of sinc(z) in the complex plane

    Homework Statement z=x+iy; f(z)=sin(z)/z find f'(z) and the maximal region in which f(z) is analytic. Homework Equations The sinc function is analytic everywhere. The Attempt at a Solution Writing f(z) as (z^{-1})sin(z) and differentiating with respect to z using the chain rule I get...
  41. T

    How Do You Plot Complex Numbers on a Plane?

    Homework Statement Graph the following numbers on a complex plane. A) 3-2i B)-4 C)-2+i D)-3i D)-1-4i Can smomeone help me how to get started on this question? I'm not sure what it wants me to do
  42. J

    Equation of ellipse: complex plane

    Homework Statement The question simply states that the focals are (0,2) and (2,-1) and I need to form an equation from it. I know that in complex form this would be |z-(0-2i)| + |z-(-2+i)| or more simply |z+2i|+|z+2-i|. Is this right?
  43. P

    MHB Finding Cells in Lattices: Exploring Complex Plane Structures

    I have the following assignment:consider the map $$|\cdot|:\mathbb{Z}[i]\longrightarrow \mathbb{N},\qquad |a+ib|:=a^2+b^2$$1) Prove that $|\alpha|<|\beta|$ iff $|\alpha|\leq |\beta|-1$ and $|\alpha|<1$ iff $\alpha=0$2) Let $\alpha,\beta\in\mathbb{Z}[i],\beta\neq 0$. Prove that the map...
  44. 4

    Equation of a line in complex plane

    All the derivation of the equation of line on complex plane uses the fact that (x,y) \in R^2 can be identified with x+iy \in C. Thus, they begin with ax+by+c = 0 then re-write x = (z+\bar{z})/2 and y = (z-\bar{z})/(2i), and substitute it into real plane line equation to get it in complex...
  45. I

    Formulation of the Complex Plane

    Hey all, More of a fundamental question, could possibly be a chicken-egg question. I understand mathematical constructs and minuta if I know from where it hailed. So my question is this, what was the motivation behind developing the complex plane? Was it theoretically developed or was it...
  46. J

    Sketching loci in the complex plane

    Homework Statement Make a sketch of the complex plane showing a typical pair of complex numbers z1 and z2 Describe the geometrical figure whose vertices are z1, z2 and z0 = a + i0. Homework Equations z2 − z1 = (z1 − a)ei2π/3 a − z2 = (z2 − z1)i2π/3 where a is a real...
  47. J

    Error function (defined on the whole complex plane) is entire

    Homework Statement The wiki page says that error function \mbox{erf}(z) = \int_{0}^{z} e^{-t^{2}} dt is entire. But I cannot find anywhere its proof. Could you give me some stcratch proof of this? Homework Equations The Attempt at a Solution I've tried to use Fundamental Theorem...
  48. S

    Mapping unit circle from one complex plane to another

    I want to show that if the complex variables ζ and z and related via the relation z = (2/ζ) + ζ then the unit circle mod(ζ) = 1 in the ζ plane maps to an ellipse in the z-plane. Then if I write z as x + iy, what is the equation for this ellipse in terms of x and y? Any help would be...
  49. W

    How Do Imaginary Components Affect Vectors in a 2D Complex Vector Space?

    I'm having difficulty understanding the nature of a plane in complex space. Specifically, I have two complex (N \times 1) vectors, \underline{u}_1 and \underline{u}_2 , which are orthonormal: \underline{u}_1^H\underline{u}_2 = 0 \\ \Vert \underline{u}_k \Vert = 1 \ \ \ (k=1,2) So...
  50. marellasunny

    What is the concept of 'one infinity' in complex analysis?

    'In the real case,we can distinguish between the limits (+infinity) and (-infinity),but in the complex case there is only one infinity'-this was given in Lars Ahlfors's book on complex analysis. Can someone explain what he means by 'one infinity'? My proffesor asked me to look into the concept...