Complex Definition and 1000 Threads

Hello, I have two questions regarding the Radius of convergence. 1. What should we do at the interval (R-eps, R) 2. It used definition to prove radius of convergence, but I am not sure if it is necessary-sufficient condition of convergence. I get that this can be a sufficient condition but not...

For a harmonic function of a complex number ##z##, ##F(z)=\frac{1}{z}##, which can be put as ##F(z)=f(z)+g(\bar{z})##and satisfies ##\partial_xg=i\partial_yg##. But this function can also be put as ##F(z)=\frac{\bar{z}}{x^2+y^2}## which does not satisfy that derivative equation! Sorry, I...

Hi! I have problems with this demonstration Let $z= x+iy , x,y \in \mathbb{R} $ then $|x|, |y| \leq{|z|} \leq{\sqrt[ ]{2}} $ , $max \{ |x|, |y| \} $

I understood the holomorphic condition this way. For a vector field F(x1, x2 . . ., xm) = <y1(x1, x2, x3 . . . , xm), y2(x1, x2, x3 . . . , xm), y3(x1, x2, x3 . . . , xm) . . . ,yn(x1, x2, x3 . . . , xm)> In a real analysis, its derivative is expressed as a Jacobian matrix because each...

We should note that we can write any complex number as $\displaystyle \begin{align*} z = r\,\mathrm{e}^{\mathrm{i}\,\theta} \end{align*}$ where $\displaystyle \begin{align*} r = \left| z \right| \end{align*}$ and $\displaystyle \begin{align*} \theta = \textrm{arg}\,\left( z \right) + 2\,\pi\,n ...

I am having trouble understanding the relationship between complex- and real-argument associated Legendre polynomials. According to Abramowitz & Stegun, EQ 8.6.6, $$P^\mu_\nu(z)=(z^2-1)^{\mu/2}\cdot\frac{d^\mu P_\nu(z)}{dz^\mu}$$ $$P^\mu_\nu(x)=(-1)^\mu(1-x^2)^{\mu/2}\cdot\frac{d^\mu...

Hi, i need advice, I'm studying complex analysis on which book would you recommend to do it on that of Ahlfors or T.W.Gamelin's book?

Homework Statement evaluate ##\int \frac{sinh(ax)}{sinh(\pi x)}## where the integral runs from 0 to infinity Homework EquationsThe Attempt at a Solution consider ##\frac{sinh(az)}{sinh(\pi z)}## Poles are at ##z= n \pi i## So I'm considering the contour integral around the closed contour from...

First let's write this number in its polar form. $\displaystyle \begin{align*} \left| z \right| &= \sqrt{\left( -2 \right) ^2 + 2^2} \\ &= \sqrt{4 + 4} \\ &= \sqrt{8} \\ &= 2\,\sqrt{2} \end{align*}$ and as the number is in Quadrant 2 $\displaystyle \begin{align*} \textrm{arg}\,\left( z...

$\displaystyle \begin{align*} z^3 + 1 &= 0 \\ z^3 &= -1 \\ z^3 &= \mathrm{e}^{ \left( 2\,n + 1 \right) \,\pi\,\mathrm{i} } \textrm{ where } n \in \mathbf{Z} \\ z &= \left[ \mathrm{e}^{\left( 2\,n + 1 \right) \, \pi \,\mathrm{i}} \right] ^{\frac{1}{3}} \\ &= \mathrm{e}^{ \frac{\left( 2\,n + 1...

My question boils down to wondering if there is a way to simplify the imaginary part of a complex-valued function composed of n factors if the real and imaginary component for each of the factors is known but the factors may take on the value of their conjugate as well. For example, is there a...

Homework Statement [/B] Suppose q(z) = z^3 − z^2 + rz + s, is a complex polynomial with 1 + i and i as zeros. Find r and s and the third complex zero. The Attempt at a Solution [/B] (z-(1+i)(z-i) = Z^2-z-1-2iz+i (Z^2-z-1-2iz+i)(z+d) = Z^3+z^2(d-1-zi)-z(d+1+2di-i)-d(1-i) Z^2 term...

Hi there! First Post :D In a recent CM module we've been looking at coupled oscillators and the role of time translational invariance in the description of such physical systems. I will present the statement that I am having trouble understanding and then continue to elaborate. In stating that...

Hi Community, I have the following problem and I am completely stuck. I really struggle to get my head around how to break down these questions into chunks that I can then apply the math to. From what I can see so far, I have a to be able calculate the surface area at any height to get the...

Hi Community, I have the following problem and I would like some help in understanding part a. So far I far I have been able to show that: 1+\frac{(e^x-e^{-x})^2}{4} = \frac{(e^x)^2-2(e^x-e^{-x})+(e^{-x})2}{4}+1 But I am unsure of how to proceed. Also any pointers on how to look at the...

Homework Statement Showing all necessary working solve the equation ##iz^2+2z-3i=0## giving your answer in the form ##x+iy## where x and y are real and exact,Homework EquationsThe Attempt at a Solution ##iz^2+2z-3i=0, z^2+(2/i)z-3=0##,using quadratic formula →##(-2/i± √8)/2 , z= √2+1/i## and...

Hey! I've been trying to tackle this problem but I'm a little lost at the moment and any references or suggestions would be greatly appreciated. Essentially the problem boils down to solving (at least) 3 coupled partial differential equations with (at least) 2 independent variables. Now the...

Homework Statement How would I go about solving 1/z=(-4+4i) The answer that I keep on getting is wrong The Attempt at a Solution [/B] What I did z=1/(-4+4i)x(-4-4i)/(-4-4i) z=(-4-4i)/(16+16i-16i-16i^2) z=(-4-4i)/32 z=-1/8-i/8 This is the answer that I got however it says that it is...

Homework Statement evaluate sinx/x^4 over the unit circle Homework Equations Cauchys Residue theorem ##sinz=1/(2i)(z+1/z)## The Attempt at a Solution So we have a branch point at z=0 but its of order 4 so I can't see any direct way of using Cauchys residue theorem. I've tried changing the...

Hey, This is my first post so I am hoping to do everything right :-) I do not understand the physical meaning of a complex wavenumber. I understand that, with a general approach u(x,t) = Re(A*[e][(i(kx-omega*t)]) and a complex wavenuber that the wave is decaying exponentially with x. What...

Homework Statement What is difference in shading between Argand diagrams containing inequalities with > and ≥ signs? Example Shade the appropriate region to satisfy the inequality |z|> 5 |z|≥ 5 The Attempt at a Solution I am aware of the fact that both will have circle centered at origin...

Homework Statement whose Fourier transform is f~(p) = 1/(a2 + p2) Homework Equations f(x) = 1/√2π ∫∞-∞ eipx f~(p)The Attempt at a Solution First of all I let f(z) = eixz/(z2 + a2) and γ = γ1 + γ2 with the ϒ's parametrised by: γ1 : {z=t, -R<t<R} γ2 : {z=Reit, 0<t<π} (So a semicircle of radius...

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...

1. Give a formula for the values on m such that z^m=z z=cos(7pi/6)+i*sin(7pi/6) 2. If i use de movires i get 3. m*7pi/6=7pi/6 + k*2pi But then i get the value that k=12/7, Which is the wrong formula. The correct answer is 1+12k for k=0,1,2...

Homework Statement Assume a particle with a wave function ##\psi(x)## such that ##-\infty < x < \infty##, that move under some potential ##V(x)##. Show that: a) two wave functions with same energies can only differ by a complex phase; b) if the potential is real, then you can choose the wave...

Hi, Is there a proof that complex replacement is a valid way to solve a differential equation? I'm lacking some intuition on the idea that under any algebraic manipulations the real and imaginary parts of an expression don't influence each other. For example, if I'm given: $$p(D) x = cos(t)$$...

Homework Statement (Z^3)+3i(conjugate z) = 0 find all solutions. Homework EquationsThe Attempt at a Solution How can i isolate Z Tried factoring out z, didn't came out good: z(z^2+3i*(Conjugate z)/z) ==> Right part equal to zero. couldn't factor anymore.

Homework Statement Let ##z_1,z_2,z_3## be three complex numbers that lie on the unit circle in the complex plane, and ##z_1+z_2+z_3=0##. Show that the numbers are located at the vertices of an equilateral triangle. Homework EquationsThe Attempt at a Solution As far as I understand, I need to...

Homework Statement Verify that i2=-1 using (a+bi)(c+di) = (ac-bd)(ad+bc)i Homework Equations (a+bi)(c+di) = (ac-bd)(ad+bc)i The Attempt at a Solution I tried choosing coefficients so that it would be (i)(i) = (0 - 1)+(0+0)i = -1 so then I get i^2 = -1 But I was told that this was wrong and...

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...

Given A(2√3,1) in R^2 , rotate OA by 30° in clockwise direction and stretch the resulting vector by a factor of 6 to OB. Determine the coordinates of B in surd form using complex number technique. i try to rewrite in Euler's form and I found the modulus was √13 but the argument could not be...

Note:- All surfaces given here are frictionless. To find:- Acceleration a Relevant eqs:- F = ma Attempt at solution:- The equations I've gotten so far:- 1) N1sin(theta) = m2a 2) m1gsin(theta) = m1a1 3) m1gcos(theta) - N1 = m1a2 So far, 3 eqs, 4 unknowns. For the 4th eq, I did a2sin(theta) =...

Hello, i am kind of confused about something. What is the complex conjugate of the momentum operator? I don't mean the Hermitian adjoint, because i know that the Hermitian adjoint of the momentum operator is the momentum operator. Thanks!

Homework Statement If ##α, β, γ, δ## are four complex numbers such that ##\dfrac{γ}{δ}## is real and ##αδ - βγ ≠ 0##, then ##z = \dfrac{α + βt}{γ + δt} , t \in ℝ## represents a (A) circle (B) parabola (C) ellipse (D) straight line Homework EquationsThe Attempt at a Solution Eqn of circle is...

Homework Statement Reflection of the line ##\bar{a}z + a\bar{z} = 0## in the real axis is Homework EquationsThe Attempt at a Solution I know that a line in the complex plane is represented as ##\bar{a}z + a\bar{z} + b= 0## and that its slope ##μ = \dfrac{-a}{\bar{a}}##. I'm not sure how to do...

Homework Statement So I have been having trouble with finding the proper eigen vector for a complex eigen value for the matrix A=(-3 -5) . .....(3 1) had a little trouble with formating this matrix sorry The eigen values are -1+i√11 and -1-i√11 The Attempt at a Solution using AY-λY=0...

Homework Statement Suppose the matrix A with real entries has the complex eigenvalue λ=α+iβ, β does not equal 0. Let Y0 be an eigenvector for λ and write Y0=Y1 +iY2 , where Y1 =(x1, y1) and Y2 =(x2, y2) have real entries. Show that Y1 and Y2 are linearly independent. [Hint: Suppose they are...

This may be a simple thing but due to some reason I am not able to understand. I am not able to understand an example from Brown-Churchill book. I have provided the screenshot in the attached picture. Request help.

Homework Statement I am not able to understand an example from Brown-Churchill book. I have provided the screenshot in the attached picture. Request help. Homework Equations No The Attempt at a Solution No

Homework Statement a) The complex number ## 1-i ## is denoted by ##u##. On an argand diagram, sketch the loci representing the complex numbers ## z## satisfying the equations ## |z-u|= |z| and |z-i|=2 ## b) Find the argument of the complex numbers represented by the points of intersection of...

Homework Statement a. Represent f(x)=|x| in -2<x<2 with a complex Fourier series b. Show that the complex Fourier Series can be rearranged into a cosine series c. Take the derivative of that cosine series. What function does the resulting series represent? [/B]Homework Equations...

Prove that $\arg[(a+bi)(c+di)]=\arg(a+bi)+\arg(c+di)$.

Homework Statement Use the complex susceptibility to prove that the dot product of E and J is related to the absorption term (the imaginary part - χ'') and independent of the real part (χ'). It is also stated that in order to do is, assume monochromatic field and take the absorption time...

Hello, I am enrolled in calculus 2. Just having started a section in our textbook about integration by partial fractions, I eagerly began trying to use this integration technique wherever I could. After messing around for multiple days, I ran into this problem: ∫ 1/(x^2+1)dx I immediately...

Homework Statement In sinusoidal circuit shown in Figure 13 is known : w =10^6 1/s, R = 100 Ohm , L = 300μH , C1 = 10nF and C2= 5nF . Reactive power of coil inductance L is QL = 3kVAr , RMS value of the voltage receiver impedance Z is UZ = 100 V , and the voltage UZ phase delaying behind...

Homework Statement State, with justification, if the Fundamental Theorem of Contour Integration can be applied to the following integrals. Evaluate both integrals. \begin{eqnarray*} (i) \hspace{0.2cm} \int_\gamma \frac{1}{z} dz \\ (ii) \hspace{0.2cm} \int_\gamma \overline{z} dz \\...

Hi, I'm not sure about the the normal vector N on a complex function z(x,t) = A e^{i(\omega t + \alpha x)} My approach is that (\overline{z} beeing the conjugate of z): \Re{(\mathbf{N})} = \frac{1}{\sqrt{\frac{1}{4}(\partial x + \overline{\partial x} )^2 + \frac{1}{4}(\partial z +...

Hi! We have discussed complex numbers in class and their conjugates. From what I understand only the imaginary unit is conjugated. But I wonder if there are such things as real conjugates of complex numbers? Given the following points: $$A=(-2+i)$$ $$B=(2+3i)$$ $$C=(-4-3i)$$ $$D=(-4+i)$$ I...

Hello this is my first post here so basically I've had my best crack at all the questions all I am really after is a bit or re assurance as to my answers and if any are wrong were I have gone wrong essentially. I've tried to include all my working out were possible. 1. Homework Statement The...

Homework Statement Homework Equations Req = [ ( 1 / R1) + ( 1 / R2) +. ...]^-1 (Parallel) Req = R1 + R2 +. . . (Series ) The Attempt at a Solution I tried to simplify the circuit by spreading it out but I guess something is wrong with my simplification since I can't arrive at the correct...