Read about complex algebra | 10 Discussions | Page 1

  1. R

    Argument of a complex expression

    Problem Statement: What is the correct way of computing the argument of the following equation? Relevant Equations: I am trying to compute the argument ##\Phi## of the equation $$\frac{r-\tau\exp\left(i\varphi\right)}{1-\tau r\exp\left(i\varphi\right)} \tag{1}$$ which using Euler's equation...
  2. Santilopez10

    De Moivre theorem problem

    Homework Statement Find all $$n \in Z$$, for which $$ (\sqrt 3+i)^n = 2^{n-1} (-1+\sqrt 3 i)$$ Homework Equations $$ (a+b i)^n = |a+b i|^n e^{i n (\theta + 2 \pi k)} $$ The Attempt at a Solution First I convert everything to it`s complex exponential form: $$ 2^n e^{i n (\frac {\pi}{3}+ 2\pi...
  3. Mutatis

    Write ##5-3i## in the polar form ##re^\left(i\theta\right)##

    Homework Statement Write ##5-3i## in the polar form ##re^\left(i\theta\right)##. Homework Equations $$ |z|=\sqrt {a^2+b^2} $$ The Attempt at a Solution First I've found the absolute value of ##z##: $$ |z|=\sqrt {5^2+3^2}=\sqrt {34} $$. Next, I've found $$ \sin(\theta) = \frac {-3} {\sqrt...
  4. Safder Aree

    Contour Integration over Square, Complex Anaylsis

    Homework Statement Show that $$\int_C e^zdz = 0$$ Let C be the perimeter of the square with vertices at the points z = 0, z = 1, z = 1 +i and z = i. Homework Equations $$z = x + iy$$ The Attempt at a Solution I know that if a function is analytic/holomorphic on a domain and the contour lies...
  5. A

    Solving Schrodinger's Equation with a weak Imaginary Potential

    Homework Statement A particle of energy E moves in one dimension in a constant imaginary potential -iV where V << E. a) Find the particle's wavefunction \Psi(x,t) approximating to leading non-vanishing order in the small quantity \frac{V}{E} << 1. b) Calculate the probability current density...
  6. Rectifier

    Complex equation

    The problem I would like to solve: $$ \bar{z} = z^n $$ where ##n## is a positive integer. The attempt ## z = r e^{i \theta} \\ \\ \overline{ r e^{i \theta} } = r^n e^{i \theta n} \\ r e^{-i \theta} = r^n e^{i \theta n} ## ## r = r^n \Leftrightarrow true \ \ if \ \ n=1 \ \ or \ \ r=1## ##...
  7. K

    I Need a help in solving an equation (probably differentiation

    I am trying to find out the interference condition between tool and a part. The below attached snapshot is the equation between interference and machine feed. At dy/dx = 0, I will have max. interference, which I intend to find. Except x and y every alphanumeric character in the following...
  8. Adgorn

    B Square root of a negative number in a complex field

    Mod note: Fixed all of the radicals. The expressions inside the radical need to be surrounded with braces -- { } (This question is probably asked a lot but I could not find it so I'll just ask it myself.) Does the square root of negative numbers exist in the complex field? In other words is...
  9. Wi_N

    Z^4=-80i or general z^n=a+bi

    Homework Statement I just cant seem to get the right answer. z^4+80i=0 looking at the complex plane u see the radius=r=80 (obviously) using De Moivre extension: z^n=(r^(1/n))(cos((x/n)+k2pi/n)-isin((x/n)+k2pi/n) z1=((80)^(1/4))(cos(3pi/8)+isin(3pi/8) shouldnt this be a root? z2=...
  10. Wi_N

    Solving z^2+pz+qiz+a+bi=0

    ok having major problems. i can easily solve z^2 + pz +a+bi=0 solutions but that extra qiz is really annoying me. z^2 + 3z+4iz-1+5i=0 (z+2i)^2+3z-5+5i=0 z+2i = w, z=w-2i w=-3(w-2i)+5-5i then im not getting anything sensible for solving x and yi. what am i doing wrong?