Complex number Definition and 438 Threads

Homework Statement Of all complex numbers that fit requirement: ## |z-25i| \leq 15## find the one with the lowest argument. Homework EquationsThe Attempt at a Solution z=a + ib (a, b are real numbers) ## \sqrt{a^2 + (b-25)^2} \leq 15 \\ a^2 + (b-25)^2 \leq 225 ## The lowest possible...

Homework Statement Given n=(x + iy)/2½L and n*=(x - iy)/2½L Show that ∂/∂n = L(∂/∂x - i ∂/∂y)/2½ and ∂/∂n = L(∂/∂x + i ∂/∂y)/2½ Homework Equations ∂n Ξ ∂/∂n, ∂x Ξ ∂/∂x, as well as y. The Attempt at a Solution ∂n=(∂x + i ∂y)/2½L Apply complex conjugate on right side, ∂n=[(∂x + i ∂y)/2½L] *...

Homework Statement Solve the following equation: ## (1+a)^n=(1-a)^n## where a is complex number and n is natural number Homework Equations Euler's formula The Attempt at a Solution I've tried something like this ## (1+a)^n=(1-a)^n \\ (\frac{1+a}{1-a})^n=1 ## But i really have no idea...

Homework Statement I have the following complex numbers : -3,18 +4,19i I must put it in polar form. Homework Equations r=(a^2+b^2)^(1/2) cos x = a/r sin x = b/r The Attempt at a Solution I was able to find with cos x = a/r that the x = 127,20 But when I do it with sin x = b/r I obtain like...

Homework Statement The problem states that you need to solve the following equation (without a calculator) : z^5 = z̅ Homework Equations z=a+bi and z̅=a-bi The Attempt at a Solution So far I've tried multiplying both sides by z̅: z̅ * z^5 = |z̅|^2...

If i understand correctly, the discovery of complex numbers was linked to solving real number problems, s.a. finding square roots of negative numbers. In other words, at first there was a problem that was formulated using real numbers only that had no real number solutions, which lead to...

The transformation T maps the plane onto itself by multiplication by a complex number. That is, there is a complex number C=a+ib such that for any point P(x,y), T(P) is the point corresponding to the complex number C⋅P. For a particular complex number C the transformation T takes the smaller...

Homework Statement The problem is to sketch lines of constant u and v in the image plane for the function Log[(z+1)/(z-1)]. Homework Equations z=x+iy The Attempt at a Solution In order to do this I have to get the expression into u+iv form, so that I can read off and manipulate the u and v...

Is correct to afirm that a solution of a quadratic equation or is a complex number or is a split complex number?

Homework Statement This is not for a mathematics unit but is part of an electrical question I'm trying to solve but I cannot solve this equation. The complex numbers Zp and Zr are both real and imaginary, whereas Xm is purely imaginary. Homework Equations Zp = (Xm*Zr)/(Xm+Zr) Zp =...

If I define the complex number z = r exp(i θ) how z = uv, so, how to express u and v in terms of r and θ? u(r, θ) = ? v(r, θ) = ? And the inverse too: r(u, v) = ? θ(u, v) = ?

I was me asking why the complex numbers are defined how z = x + i y !? Is this definition the better definition or was chosen by chance? In mathematics, some things are defined by chance, for example: 0 is the multiplicative neutral element and your multiplicative inverse (0-) is the ∞. But, 1...

I am interested to find the length shown in red in the attached figure. I want this length as a function of d (shown in blue) and the angle θ. Then I will integrate this length to dθ from 0 to π/2. Firstly, I used the law of the triangle to determine the length s which when subtracted from the...

As you can see, it says that -110 (-1 to the tenth is just -1), multiplied by i, is somehow i. Everywhere I have looked, -1 times i is negative i, but this problem disagrees. Am I missing something?

Hi - in an example, I can't follow the working from one of the steps to the next, the 2 steps are: $... \sqrt{\frac{1}{2}\left(1-i\right)} = \sqrt{\frac{1}{\sqrt{2}}{e^{-i(\frac{\pi}{4}-2n\pi)}}}$ I can see they equate $ \frac{1-i}{\sqrt{2}} = e^{-i(\frac{\pi}{4}-2n\pi)}$, and I can see the $...

Hi everyone, I have a dispersive wave packet of the form: ##\frac{1}{\sqrt{D^2 + 2i \frac{ct}{k_0}}} e^{-y^2/(D^2+2i\frac{ct}{k_0})}## The textbook says that the enlargement of the package, on the y direction, is: ##L=\frac{1}{D}\sqrt{D^4+4\left(\frac{ct}{k_0}\right)^2} ## However I have some...

I'm working through some examples in a textbook but i am unable to get the desired answer on my calculator, i keep getting math error and various other results which are not the answer I'm looking for. What i have is: √ 62.9∠88.2 / 0.00165∠72.3 Please could someone tell me what answer you get...

i have a a little problem in fortan90 i just wanted to know how to input a complex number ( input real and img part alone ) all i want to do is to make a simple program about DeMoivres Theorem i have been around in google all i know how to declare a argument as complex complex a then how to...

When I solve a quadratic equation I need to find a Discriminant. If D>0 I have no problem. I can find x1 and x2. And when I draw a parabola I can see the x1 and x2 on a X-line. But when D<0 I don't understand where I can find x1 and x2 on a plot of function. For example for 5x2+2x+1=0 I...

Homework Statement Good day, I've been have having difficulties finding the roots of this: Find the roots of 3ix^2 + 6x - i = 0 where i = complex number i = sqrt(-1) Homework Equations quadratic formula (apologies for the large image) The Attempt at a Solution...

Homework Statement How would Re(z)<0 be graphed? Homework Equations Re(z) is the real part of z The Attempt at a Solution It looks similar to y>x, but only shaded in the third quadrant, how can this be explained? not relevant anymore

we know Complex number is consists of a real & imaginary number so if a+ib is a complex number which one is real part, imaginary part, real number and imaginary number? I think "a" real and "ib" imaginary part...pleasez help me if Iam wrong ...

Homework Statement Decompose x5 - 1 into the product of 3 polynomials with real coefficients, using roots of unity. Homework Equations As far as I know, for xn = 1 for all n ∈ ℤ, there exist n distinct roots. The Attempt at a Solution [/B] So, let ω = e2πi/5. I can therefore find all the 5th...

Hey all. I am giving Panhellenic exams this year, and part of my preparation comes from solving previous' years questions. This problem was a complex number based one, in the 2013 Panhellenic exams. It goes as follows: Consider 3 complex numbers, α0,α1 and α2 which belong in the line given by...

Homework Statement Find the solutions to z^{\frac{3}{4}}=\sqrt{6}+\sqrt{2}i Homework Equations de Moivre's theorem The Attempt at a Solution z^{\frac{3}{4}}=2\sqrt{2}e^{\frac{\pi i}{6}}=2\sqrt{2}e^{\frac{\pi i}{6}+2k\pi}=2\sqrt{2}e^{\frac{\pi +12k\pi}{6}i} z=4e^{\frac{4}{3}{\frac{\pi...

Homework Statement Prove that \lim_{z\rightarrow z_0} Re\hspace{1mm}z = Re\hspace{1mm} z_0 Homework Equations It is specifically mentioned in the text that the epsilon-delta relation should be used, |f(z)-\omega_0| < \epsilon\hspace{3mm}\text{whenever}\hspace{3mm}0<|z-z_0|<\delta . Where...

Homework Statement Find d^2/dx^2 and both complex number forms for the complex number equation (1+icos(x))/(1-icos(y))[/B] Homework Equations 1. z=a+bi 2. re^itheta The Attempt at a Solution I have multiplied both sides by 1+icosy and gotten as far as (1+icosx+icosy-cosxcosy)/(1+cos^2y) but...

Homework Statement The Attempt at a Solution -1 < (z-w) /(1-z*w) < 1 [/B] Hi can give clue. I am clueless

Given that 1/z = 1/2 + 1/(iw) + 1/(i+1)w Express the real and imaginary parts of z in term of w.Someone please help me to solve it.. THANK YOU!

Please, do not give me answers to these questions, I just want to have some hints/tips to point me in the correct direction, or if you're so inclined, you can use another similar/relevant to help me understand this concept. Thanks! Homework Statement a.) [/B]Given that z/(z + 2) = 2-i, z ∈ ℂ...

What's the set \{ z \in \mathbb{C}| |z|^2 \geq z+ \bar{z} \}? I've set z=a+ib and found a^2 + b^2 \geq 2a \Rightarrow b^2 \geq a(2-a) I'm not sure how to interpret this geometrically ie what it looks like? I suppose it is the set of vectors whose length is bigger than twice their real part. I...

Hello, I have the point ##-1 + 2i##, for which I am asked to find the modulus and argument. The modulus was simple enough, but I am having difficulty finding the angle. The point is located in the 4th quadrant, and so I need to make certain that I calculate an angle in the range ##(\frac{3...

Homework Statement a) Find the modulus and argument of 6^(1/2) + 2^(1/2)i b) Solve the equation z^(3/4) = 6^(1/2) + 2^(1/2)i Homework Equations The Attempt at a Solution For part a) i used Pythagoras to find the modulus. ( (6^(1/2))^2 + (2^(1/2))^2 )^(1/2) = (6 + 2)^(1/2)...

Homework Statement Hi guys, I have no idea what this question wants me to do. Any clarification would be appreciated. Let z0, z1, z2, z3 and z4 be the solutions that you obtained. Use the factorization z5+1= (z - z0) (z - z1) (z - z2) (z - z3) (z - z4) to determine the complex...

I'm trying to get straight some basic complex number fundamentals. First we have z=x+iy. Ok, my question is what does the z in the equation represent? Does it represent a "point" on the complex plane? Does it represent a vector from the origin to where the x and iy values add? Does it...

Homework Statement Calculate argument of complex number -1-\sqrt{3}i Homework Equations The Attempt at a Solution The argument of this is -120 degrees but why couldn't we as well say it's 240 degrees? Since going 240 degrees will go to the same point as -120 degrees. Why is this false?

Evaluate the following logarithms, expressing the answers in rectangular form a. $\ln1$, $Ln1$ b. $\ln(3-j4)$, $Ln(3-j4)$ I know that the log of a complex number z is given as $\ln z=\ln|z|+argz$ but I still don't know how to use this fact to solve the problems above. I'm having a hard...

I'm trying to solve this problem and got stuck. Find the roots of $\sqrt{-j}$ converting $0-j$ into polar form $r=\sqrt{0^2-1^2}=1$ $\theta=\tan^{-1}\left(\frac{-1}{0}\right)$ I got stuck on this part. please help.

1. The problem. Given that z= 3-4i Show that z^2 = 3-4i Hence or otherwise find the roots of the equation (z+i)^2=3-4i 2. My attempt. The first part of the problem is strait forward z^2= (2-i)(2-i) then expand to get the desired result. Now the second part (z+i)^2=3-4i. Becomes z^2+...

I just wanted to check something. If I have a complex number of the form a = C * \exp(i \phi) where C is some non-complex scalar constant. Then the phase of this complex number is simply \phi. Is that correct?

Hello. I am not confident about this question. I think I have to use cauchy integral formula. But before that, I should decompose the fraction, right? Or is there a simpler way to do it?　A friend told me that each contour only had one pole interior to it so he just used the Cauchy integral...

Hello. I am not confident about this question. I think I have to use cauchy integral formula. But before that, I should decompose the fraction, right? Or is there a simpler way to do it?　A friend told me that each contour only had one pole interior to it so he just used the Cauchy integral...

Hello. I am stuck at the third point, that is from 1+i to i. I asked someone to show me his answer but that part of his is different from mine. Is his solution correct? Here it is: (i) z = 0 to 1 via z(t) = t with t in [0, 1]: ∫c1 Re(z^2) dz = ∫(t = 0 to 1) Re(t^2) * 1 dt = ∫(t = 0 to 1)...

Hello. I am stuck at the third point, that is from 1+i to i. I asked someone to show me his answer but that part of his is different from mine. Is his solution correct? Here it is: (i) z = 0 to 1 via z(t) = t with t in [0, 1]: ∫c1 Re(z^2) dz = ∫(t = 0 to 1) Re(t^2) * 1 dt = ∫(t = 0 to 1) t^2 dt...

can anyone explain how ro make the working above the red circle to the working in the red circle? why the author do this?

Homework Statement fnd 1+i\sqrt{3}/1+i knowing sin ∏/12 cos ∏/12 Homework Equations The Attempt at a Solution Our teacher did not really teached me how to do it...

Homework Statement I have to find ##\tan^{-1}(2i)##. Homework Equations The Attempt at a Solution So far I have ##\tan^{-1}(2i)=z\iff tan z= 2i\iff \dfrac{sin z}{cos z}=2i ##. From here I get that ##-3=e^{-2zi}##. I do no know how to take it further to get ##z=i\dfrac{\ln...

Problem: If $z$ is a complex number such that $$\arg(z(1+\overline{z}))+\arg\left(\frac{|z|^2}{z-|z|^2}\right)=0$$ then A)$\arg(\overline{z})=-\pi/2$ B)$\arg(z)=\pi/4$ C)$|\overline{z}|<1$ D)$\ln\left(\frac{1}{|z|}\right)\in (-\infty,\infty)$ Attempt: From the fact that...

In the context of complex number, how to prove that 1. log i +log(-1+i) \neq log i(-1+i) 2. log i^2 =2log i

in the following i will demonstrate a 'proof' that 1+1=0 1+1=√1 +1 =√(-1)(-1) +1 = (√(-1))(√(-1)) +1 = (i)(i) +1 = i2 +1 = -1 + 1 = 0 I know I'm not the first to come up with this 'proof', and i have been told that the problem lies with splitting the...