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.
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...
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...
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...
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...
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...
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:
$$g(u)=\int f(\alpha)e^{\alpha^{*}u-\alpha u^{*}}d^{2}\alpha, (3.94a)$$...
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...
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...
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...
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...
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...
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 '...
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...
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...
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)=...
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...
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...
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...
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...
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...
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...
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...
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|...
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...
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...
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...
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...
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...
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...
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}...
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...
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??
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...
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...
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...
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...
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...
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:
$$|z|^2 - 2Re(\bar{a}z) + |a|^2 = r^2 $$
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...
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
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?
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...
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...
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...
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...
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...
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...
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...
'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...