Complex Definition and 1000 Threads

In Lecture 7, Prof. Arthur Mattuck (MIT OCW 18.03) taught that the following equation $$ y’ +ky = k \cos(\omega t)$$ can be solved by replacing cos⁡(ωt) by ##e^{\omega t}## and, rewriting thus, $$ \tilde{y’} + k\tilde{y}= ke^{i \omega t} $$ Where ##\tilde{y} = y_1 + i y_2##. And the solution of...

Consider the equation of motion for a simple harmonic oscillator: ##m\ddot {x}(t)=-kx(t).## The solutions are ##x(t)=Ae^{i\omega t}+Be^{-i\omega t},## where ##\omega=\sqrt{\frac{k}{m}}##, and constants ##A## and ##B##. Physically, what does it mean for a solution to be complex? Is it only the...

I know of two reasonable ways to represent a complex fraction: \dfrac{ \left ( \dfrac{a}{b} \right ) }{ \left ( \dfrac{c}{d} \right ) } ##\dfrac{ \left ( \dfrac{a}{b} \right ) }{ \left ( \dfrac{c}{d} \right ) }## and \dfrac{ ^a / _b }{ ^c / _d } ##\dfrac{ ^a / _b }{ ^c / _d }## What I am...

Hello, I'm posting here since what follows is not about homework, but constitutes a personal research which underlies some more general questions. As with the infamous "casus irreducibilis" (i.e. finding the real roots of a cubic function sometimes requires intermediate calculations with...

See the work below: I feel like it that I did it correctly. I feel like I skip a step in my induction. Please point any errors.

I am not sure why criss-cross approach would work here, but it seems to get the answer. What would be the reason why we could use this approach? $$\frac {z-1} {z+1} = ni$$ $$\implies \frac {z-1} {z+1} = \frac {ni} {1}$$ $$\implies {(z-1)} \times 1= {ni} \times {(z+1)}$$

This pic is from an older text called Tables of Higher Functions (interestingly both in German first then English second) that I jumped at buying from some niche bookstore for $40. Was this hand drawn? I think I’ve seen was it that mathegraphix or something like that linked by @fresh_42...

Hello. I am reading about state equations from a physics textbook, Physics by Frederick J. Keller, W. Edward Gettys, Malcolm j. Skove (Volume I). I don't understand some parts but since I have the Turkish translation of the book I must translate it as good and clear as possible. "State...

I apologize in advance for my English. I want to know if my solution is correct. :) To verify that for every complex number z, the numbers z + z¯ and z × z¯ are real. My solution: z = a + bi z¯ = a - bi z + z¯ = a + bi + a - bi = 2a ∈ R z × z¯ = (a + bi) × (a - bi) = a^2 + b^2 ∈ R

...and is it ever useful? An arbitrary complex number has the form ##z = a + bi## where ##a, b \in \mathbb{R}## and the dot product of two arbitrary vectors ##\vec{v} = \binom{v_1}{v_2}## and equivalently for vector ##\vec{w}## is ##\vec{v} \cdot \vec{w} = v_1 w_1 + v_2 w_3## Then the ##z## may...

Let z = x + iy $$\arg \left(\frac{1+z^2}{1 + \bar z^{2}}\right)=\arg (1+z^2) - \arg (1 + \bar z^{2})$$ $$=\arg (1+x^2+i2xy-y^2)-\arg(1+x^2-i2xy+y^2)$$ Then I stuck. I also tried: $$\frac{1+z^2}{1 + \bar z^{2}}=\frac{1+x^2+i2xy-y^2}{1+x^2-i2xy+y^2}$$ But also stuck How to do this question...

[Thread moved from the technical forums to the schoolwork forums by the Mentors] Hi i have this assignment for homework: There is only one battery for the circuit, E=10V, R=4 Ohms and L=1H it asks me to find the time constant of the circuit. i know that a time constant in a RL circuit is t=L/R...

I am stuck deriving the gauge field produced in curved spacetime for a complex scalar field. If the underlying spacetime changes, I would assume it would change the normal Lagrangian and the gauge field in the same way, so at first guess I would say the gauge field remains unchanged. If there...

Summary:: summation of the components of a complex vector Hi, In my textbook I have ##\widetilde{\vec{E_t}} = (\widetilde{\vec{E_i}} \cdot \hat{e_p}) \hat{e_p}## ##\widetilde{\vec{E_t}} = \sum_j( (\widetilde{\vec{E_{ij}}} \cdot {e_{p_j}}*) \hat{e_p}## For ##\hat{e_p} = \hat{x}##...

(z-3)3=-8, solve for z. I'm new to complex numbers, so I'm stuck on this basic problem: how do you find all real and non-real solutions in the equality, (z-3)^3=-8? Thanks a bunch.

I need help with a derivation of an equation given in a journal paper. My question is related to the third paragraph of this paper: https://doi.org/10.1007/BF00619826. Although it is about fibre coupling my problem is purely mathematical. It is about solving a complex double integral. The...

ok here i have, ##x^2+y^2-5x=0## ##-y= 2## I end up with the quadratic equation, ##x^2-5x+4=0## Finally giving us, ##z=4-2i## and ##z=1-2i##

For part (a), ##z##=##\dfrac {3+i}{3-i}## ⋅##\dfrac {3+i}{3+i}## ##z##=##\dfrac {4}{5}##+##\dfrac {3}{5}i## part (b) no problem as long as one understands the argand plane... For part (c) Modulus of ##z=1## and Modulus of ##z-z^*##=##\frac{6}{5}i##

Find the problem here; ( i do not have the solutions...i seek alternative ways of doing the problems) ok, i let ##z=x+iy## and ##z^*= x-iy##... i ended up with the simultaneous equation; ##2x+y=4## ##x+2y=-1## ##x=1## and ##y=2## therefore our complex number is ##z=1+2i##

Find the values of tan-1(1+2i). We can use the fact: tan-1z = (i/2)log((i+z)/(i-z)). Then with substitutions we have (i/2)log((1+3i)/(-i-1)). Then I think the next step would be (i/2)(log(1+3i)-log(-1-i)). Do we then just proceed to solve log(1+3i) and log(-1-i)? I'm just a little confused...

I took the equation and rewrote it as: e(i+1)(log(1-i)-log(√2)). So I worked on it in sections meaning e(i+1) and then log(1-i). For e(i+1) I got eie1 and used Euler's formula for ei to get: e1(cos(1)+isin(1)). And then for log(1-i) I got ln√2 + i(-(π/4)+2kπ). Do I just bring them...

This is a pedagogical /time management / bandwidth / tradeoff question, no argument that learning the complex exponential derivation is valuable, but is it a good strategy for preparing for first year Calculus? my 16YO son is taking AP precalc and AP calc next year and doing well, but struggled...

Some more details on the system are that L1 is very long (close to 100ft) and L2 is close to 30ft (the vertical pipes). The piping is all schedule 40 1/2" OD. Moment of inertia is roughly 10^-8. Components are about 2kg each. The distance of the pipes horizontally is small (around 2ft). Pressure...

For the electrical resistance ##R## of an ideal wire, we all know the formula ##R=\rho * \frac{l}{A}##. However this is only valid for a cylinder with constant cross sectional area ##A##. In a cone the cross section area is reduced over its height (or length ##l##). What is a good general...

First of all I am not sure which type of singularity is ##z=0##? \ln\frac{\sqrt{z^2+1}}{z}=\ln (1+\frac{1}{z^2})^{\frac{1}{2}}=\frac{1}{2}\ln (1+\frac{1}{z^2})=\frac{1}{2}\sum^{\infty}_{n=0}(-1)^{n}\frac{(\frac{1}{z^2})^{n+1}}{n+1} It looks like that ##Res[f(z),z=0]=0##

Find the question below; note that no solution is provided for this question. My approach; Find part of my sketch here; * My diagram may not be accurate..i just noted that, ##OP## takes smallest value of ##12## when ##|z+5|=|z-5|## i.e at the end of its minor axis and greatest value ##13##...

This is also pretty easy, ##z^5=(a+bi)^5## ##(a+bi)^5= a^5+\dfrac {5a^4bi}{1!}+\dfrac {20a^3(bi)^2}{2!}+\dfrac {60a^2(bi)^3}{3!}+\dfrac {120a(bi)^4}{4!}+\dfrac {120(bi)^5}{5!}## ##(a+bi)^5=a^5+5a^4bi-10a^3b^2-10a^2b^3i+5ab^4+b^5i## ##\bigl(\Re (z))=a^5-10a^3b^2+5ab^4## ##\bigl(\Im (z))=...

This is pretty straightforward, Let ##z=a+bi## ## \bigl(\Re (z))=a, \bigl(\Im (z))=b## ##zz^*=(a+bi)(a-bi)=a^2+b^2 =\bigl(\Re (z))^2+\bigl(\Im (z))^2## Any other approach? this are pretty simple questions ...all the same its good to explore different perspective on the same...

Easy questions, just a lot of computation... $$\frac {z}{z^*}=\frac {a+bi}{a-bi} ×\frac {a+bi}{a+bi}$$ $$c+di=\frac {a^2-b^2}{a^2+b^2}+\frac {2abi}{a^2+b^2}$$ $$⇒c^2= \frac {a^4-2a^2b^2+b^4}{(a^2+b^2)^2}$$ $$⇒d^2= \frac {4a^2b^2}{(a^2+b^2)^2}$$ Therefore...

$$(1+i)z+(2-i)w=3+4i$$ $$iz+(3+i)w=-1+5i$$ ok, multiplying the first equation by##(1-i)## and the second equation by ##i##, we get, $$2z+(1-3i)w=7+i$$ $$-z+(-1+3i)w=-5-i$$ adding the two equations, we get ##z=2##, We know that, $$iz+(3+i)w=-1+5i$$ $$⇒2i+(3+i)w=-1+5i$$...

Let: ##z=x+iy## $$z+\frac 1 z =1+2i$$ $$x+iy +\frac{1}{x+iy}=1+2i$$ $$x+iy+\frac{1}{x+iy} . \frac{x-iy}{x-iy}=1+2i$$ $$x+iy+\frac{x-iy}{x^2+y^2}=1+2i$$ $$\frac{x^3+xy^2+x+i(x^2y+y^3-y)}{x^2+y^2}=1+2i$$ So: $$\frac{x^3+xy^2+x}{x^2+y^2}=1$$ $$x^3+xy^2+x=x^2+y^2$$ and...

According to e.g. Keith Conrad (https://kconrad.math.uconn.edu/blurbs/ choose Complexification) If W is a vector in the vector space R2, then the complexification of R2, labelled R2(c), is a vector space W⊕W, elements of which are pairs (W,W) that satisfy the multiplication rule for complex...

For a complex null tetrad ##(\boldsymbol{m}, \overline{\boldsymbol{m}}, \boldsymbol{l}, \boldsymbol{k})##, how to arrive at formulae (3.14), (3.15) and (3.17)? The equation (3.16) is clear as is. (I checked already that they work i.e. that ##\boldsymbol{e}_a' \cdot \boldsymbol{e}_b' = 2m'_{(a}...

$$\int_{0}^{2\pi } (1+2cost)^{n}cos(nt) dt$$ $$e^{it} = z, izdt = dz$$ $$\oint (1+e^{it}+e^{-it})^{n}\frac{e^{nit}+e^{-nit}}{2} \frac{dz}{iz} = \oint (1+z+z^{-1})^{n}\frac{z^{n}+z^{-n}}{2} \frac{dz}{iz}$$ $$\oint (z+z^{2}+1)^{n}\frac{z^{2n}+1}{z^{2n+1}} \frac{dz}{2i} = \pi Res = \pi...

Can someone please tell me why this is true? This isn't exactly the De Moivre's theorem. Thank you.

This isn't a homework problem, but a more general question. Let ##f## be a function with two singular points ##r## and its complex conjugate ##r^*##. let $$f=\frac{g}{z-r} \quad \text{and assume} \quad g(r)\neq 0$$ so ##r## is a simple pole of ##f##. we have conjugates that are singular...

I went ahead and tried to prove by induction but I got stuck at the base case for ## N =1 ## ( in my course we don't define ## 0 ## as natural so that's why I started from ## N = 1 ## ) which gives ## \sum_{k=0}^1 z_k = 1 + z = 1+ a + ib ## . I need to show that this is equal to ## \frac{1-...

There is a box with 2324784 bullets. Balls are divided into 5 groop and there is a definition for each groop and how many balls and what is the probability of getting each groop. The game with the return of the balls List the probability and quantity groop 1 906192 0.389796213 groop 2 1006880...

My attempt: Let us put $\frac{1}{i+t} = \frac{1+e^{is}}{2i} \Rightarrow \frac{2i}{i+t} -1= e^{is}$ So, $\cos{s}- i\sin{s}= \frac{2i}{i+t} - 1,\Rightarrow \cos^2{(s)} - \sin^2{(s)} = \frac{-2}{(i+t)^2} +1 -\frac{4i}{i+t}$ After doing some more mathematical computations, I got $\cos{s}=...

This is the problem; Note that i am conversant with the above steps shown in the solution, having said that i realized that we could also borrow from the understanding of gradient and straight lines in finding the distance ##CD##... it follows that the equation of ##BA= -1.5x-0.5##, implying...

Can anyone recommend a book in which complex eigenvalue problems are treated? I mean the FEM analysis and the theory behind it. These are eigenvalue problems which include damping. I think that it is used for composite materials and/or airplane engineering (maybe wing fluttering?).

Drawing: I decided to attempt to approach this as several collisions. So we can start with this: Object 1-Object 2 This collision is elastic, so we know that ##P_i = P_f##. We also know ##K_i = K_f##. So, $$mv_{1i} = mv_{1f} + mv{2_f}$$ $$1 = v_{1f} + v_{2f}$$ $$v_{2f} = 1 - v_{1f}$$ and...

Title says it all. I am interested in studying the physics complex systems and nonlinear physics, however i find it very hard to find a good program as it seems this area of study is not the most mainstream of them. I found out about Max Planck but still want to know if there are other strong...

Hi, I have this problem and its solution but i know what right size is, but i don't understand what left size (delta*w(z, z)) is equal to

Hi Here is my attempt at a solution for problems 1) and 2) that can be found within the summary. Problem 1) a = 3-2i b= -6-4i c= 4+ 6i d= -4+3i Now, to calculate each vector modulus, I applied the following formula: $$\left| Vector modulus \right| = \sqrt {(a^2 + b^2) }$$ where a = real part...

(e^(i*theta))^2 = (sin(theta)+i*cos(theta))^2 = cos(theta)^2 - sin(theta)^2 + 2*i*sin(theta)*cos(theta), so the real part would be: cos(theta)^2 - sin(theta)^2, and the imaginary part would be: 2*i*sin(theta)*cos(theta). But then I don't know where to start with the modulus or the argument?

Help! I am confused with my assignment. it was about complex acid base titration. the analyte was citric acid and the titrant was a base. To find it, I need to search for C1V1=C2V2, however during the balanced equation, there is 3 mol of my base that will be reacting to 1 acid. Will the ratio of...

Hi, I'm trying to find the residue of $$f(z) = \frac{z^2}{(z^2 + a^2)^2}$$ Since I have 2 singularities which are double poles. I'm using this formula $$Res f(± ia) = \lim_{z\to\ \pm ia}(\frac{1}{(2-1)!} \frac{d}{dz}(\frac{(z \pm a)^2 z^2}{(z^2 + a^2)^2}) )$$ then, $$\lim_{z\to\ \pm ia}...

Hey everyone! I got stuck with one of my homework questions. I don't 100% understand the question, let alone how I should get started with the problem. The picture shows the whole problem, but I think I managed doing the a and b parts, just got stuck with c. How do I find the largest region in...