Complex numbers Definition and 724 Threads

Homework Statement Find the sum of the series \displaystyle S_1=1 + \frac{x^3}{3!}+\frac{x^6}{6!}+\,\dots Can't seem to get the bit above to show up nicely, should be 1+x^3/3! +x^6/6! +... Sorry! Homework Equations In a prior part of the question I had to find the complex roots of z3-1=0...

We have a,b,c different complex numbers so (a+b)^3 = (b+c)^3 = (c+a)^3 Show that a^3 = b^3 = c^3 From the first equality I reached a^3 - c^3 + 3b(a-c)(a+b+c) = 0 How a is different from c => a-c is different from 0 How do I show that a^3 - c^3 = 0?

I have solved the roots of a quadratic equation and want to "test" them by putting them back in for x. I am having a problem with the x^2 term. Of the two roots, I'm only trying so far the positive square root case. I am trying to avoid writing all my work out since that would be hell and I also...

Ok there is no way I am writing out all the work of this question using a keyboard, and my scanner chose today not to work ( yes, it chose to be an idiot and not work *VERY* grumpy face) so I can't upload a picture of my work. If I were to type out the following it think it would be very...

say we had a complex number w^4 such that w^4 = -8 +i8\sqrt{3} so w = 2(cos(\frac{\pi}{6} + \frac{k\pi}{2}) + isin(\frac{\pi}{6} + \frac{k\pi}{2})) where k is an integer in a question I was asked to find the roots of w, as there will be 4 my first assumption is that the roots would be...

Hello all, I was wondering if there was a way of solving a matrix on a TI-36X Pro that has complex numbers in it. Every time I try, it just says "invalid data type". Is there any way of getting around this? Thanks! Robert

Solve for complex number, $z$:\[\text{arg}\left( \frac{3z-6-3i}{2z-8-6i}\right)=\frac{\pi}{4}\]and \[|z-3+i|=3\]The problem I am facing is that when I substitute $z=x+iy$, the equations become extremely complicated. There has to be another tricky method which I am not able to figure out.

Homework Statement z^2 = a + bi a = real number b = real number find all the solutions for z Homework Equations The Attempt at a Solution (x+y)^2 = a + bi ?

Homework Statement Evaluate ∫^{∏}_{0}e(1+i)xdx Homework Equations I know that the Real part of this is -(1+e∏)/2 and the Imaginary part is (1+e∏)/2, but I can't get the right solution. I tried using u-substitution to create something that looked like ∫^{∏}_{0}((eu)/(1+i))du but I...

The answer is a number... Work done so far [FONT=Helvetica] 3^3 * 3^( i8pi/ln(3) ) = 27 * (3^ i8pi - 3^ln(3) ) = 27 *( 3^25.1i -3.34) [FONT=Helvetica]= 3^(3+ i25.1) - 90.26 If this is correct how can I convert this to a number...

Homework Statement Ok so before I begin I've done my share of research and have had no luck as people seem to be moving to calculators for answers these days. I know my rectangular and polar conversions in AC but I haven’t done too much work with complex numbers. I basically have two...

Homework Statement Formulate a conjecture for the equation (z^3)-1=0, (z^4)-1=0 (z^5)-1=0 and prove it. Homework Equations r^n(cosnθ + isinnθ) The Attempt at a Solution Well my conjecture is that 2pi/n and 2pi/n + pi are possible values. I'm a bit iffy on how to word it. don't...

Can someone please prove the formulas: The real part of z= 1/2(z+z*) And the imaginary part of z= 1/2i(z-z*) I can't understand why it is like this. Could someone please give me the proof?

How can I use complex numbers to evaluate an integral? For instance I'm reading a book on complex numbers and it says that to evaluate the integral from 0 to pi { e^2x cos 4x dx }, I must take the real part of the integral from 0 to pi { e^((2 + 4i)x) dx}. It totally skips how you do that. I...

Homework Statement Find the number of complex numbers satisfying |z|=z+1+2i Homework Equations The Attempt at a Solution Let z=x+iy |x+iy| = (x+1)+i(2+y) Squaring and taking modulus |\sqrt{x^{2}+y^{2}}|^{2} = |(x+1)+i(2+y)|^{2} x^{2}+y^{2} = (x+1)^{2}+(2+y)^{2} Rearranging and...

Not homework as such, just need some clarification. When finding \alpha do you have to take the signs into account when finding tan^{-1} x/a. Does it matter if a or x are negative? Next question is about quadrants 1: \theta = \alpha 2: \theta = \pi - \alpha 3: \theta = -\pi -...

sketch the line described by the equation |z - u| = |z| u = -1 + j√3 I don't understand how to do this. I'm not sure of what z is exactly. Can anybody explain to me what it is?

so I have z1 = 3 + j, z2 = -5 + 5j, z3 = z1+z2 = -2 + 6j the question is, show that angle(z1) and angle(z1 + z2) differ by an integer multiple of pi/2. I tried doing it this way arctan(z1/z3), but then I always end up with a number that doesn't work. I know that arctan(x) cannot equal...

Homework Statement 1. z^6=(64,0) 2. z^4=(3,4) Homework Equations These are expanded out into Real and Imaginary components (treat them seperate): 1. REAL (EQ 1) - x^6-15x^4y^2+15x^2y^4-y^6=64 IMAG (EQ 2) - 6x^5y-20x^3y^3+6xy^5=0 From here, you basically solve these for all six...

Homework Statement Evaluate (find the real and complex components) of the following numbers, in either rectangular or polar form: \sqrt{\frac{1+j}{4-8j}} Homework Equations The Attempt at a Solution I get to here and am not sure where to go from here \sqrt{-1/20+3/20j}...

Homework Statement u = -1 + j\sqrt{3} v = \sqrt{3} - j Let a be a real scaling factor. Determine the value(s) of a such that |u-a/v| = 2\sqrt{2} Homework Equations The equation above is the only relevant equation. The Attempt at a Solution I have converted the cartesian...

Homework Statement Evaluate (find the real and complex components) of the following complex numbers, in either rectangular or polar form: z_{1} = \frac{j(3-j4)^{*}}{(-1+6j)(2+j)^{2}} Homework Equations e^{jθ} = cosθ + j sinθ The Attempt at a Solution I sadly don't even know where to begin...

Homework Statement Let z and w be two complex numbers such that zw = 0. Show either z = 0 or w = 0. Hint: try working in exponential polar form Homework Equations z = reiθ The Attempt at a Solution I have absolutely no clue.

Hi all, I am a bit rusty and have hit a snag with decomposition of partial fractions, I am taking an Engineering course dealing with Laplace transforms. The example is: F(s)= 3 / s(s2+2s+5) Now I get that there are complex roots in the denominator and that there are conjugate complex...

1- is there any complex number, x ,such that x^x=i? 2- (-1)^(\sqrt{2})=?

Mod note: These posts are orginally from the thread: https://www.physicsforums.com/showthread.php?t=626545 The square root is not defined everywhere, at least not as a function, but as a multifunction, since every complex number has two square roots. I mean, the expression z1/2 is ambiguous...

I'm guessing that if z\in \mathbb C, then we have \left| z^{-1/2} \right|^2 = |z^{-1}| = |z|^{-1} = \frac{1}{|z|}. Proving this seems to be a real headache though. Is there a quick/easy way to do this?

Pretend that you are expaining the following to someone who knows nothing about complex numbers and within a universe where complex numbers have not been invented. In examining the function f(x) = \frac{1}{1 + x^2} we can derive the series expansion \sum_{n=0}^\infty (-1)^n x^{2n} We...

Homework Statement The Attempt at a Solution do you see where it sees sqrt((1-i)(1+i)+9)? It should be (1+i)(1+i) Why isn't it?

Are there any numbers in mathematics that are not complex numbers?

Homework Statement The Attempt at a Solution I don't understand this equation. 2pi, 4pi, 6pi only = 0 when there is a sine function before it, so I don't see how the evens = 0. I don't see why the e vanishes. I also can't get the i's to vanish since one of them is in exponential...

Say we a have a sum of spin up plane wave solutions to the Dirac equation which represent the wave-function of a localized spin-up electron which is 90% likely to be found within a distance R of the origin of a spherical coordinate system. Four complex numbers at each spacetime point are needed...

I couldn't really think of a good title for this question, lol. Is it possible that a real number exists that can only be expressed in exact form when that form must includes complex numbers? For example, the equation 2 \, x^{3} - 6 \, x^{2} + 2 = 0 has the following roots x_1 =...

Homework Statement Here is the problem: (\sqrt{}6(cos(3pie/16)+i sin(3pie/16)))^4Homework Equations After 360*=2pier Do you add 2pier to the next degree which would be 30*=pie/6? The Attempt at a Solution

Homework Statement Sketch all complex numbers z which satisfy the given condition: |z-i|\geq|z-1| Homework Equations z=a+bi |z|=\sqrt{a^{2}-b^{2}} The Attempt at a Solution First I find the boundary between the regions where the inequality holds and does not hold by...

I doing some reading on why the wave-function is complex. From what I can tell, it's due to its evolution by unitary operators. But unitary operators seem to have something to do with information conservation. So I wonder if these idea have been developed somewhere in a concise fashion that...

Hello all! I'm new in the forum, and in complex numbers so I sorry for my mistakes. I have some questions about complex numbers. So can I store two values in complex number for e.g. a particle position and velocity, like xe^{i\dot{x}}? And if this works, after I get the complex number how can...

Why is the modulus of z, a complex number, |z| = √(a^2+b^2)? Why is it not |z| = √(a^2+(ib)^2)?

Homework Statement I'm trying to follow some solution to an exercise in physics and apparently e^{-im \frac{3\pi}{2}}=i^m where m \in \mathbb{Z}. I don't realize why this is true. Homework Equations Euler's formula.The Attempt at a Solution I applied Euler's formula but this is still a...

Homework Statement solve the following and determine if it is complex, real and/or imaginary x2 = 1 The Attempt at a Solution The answer is x = +-1 which I agree with but the book says that that is complex and real. How complex? Complex numbers have the form a + bi, there's no...

Homework Statement Verify that the equation (z1z2)^w=(z1^w)(z2^w) is violated for z1=z2=-1 and w=-i. Under what conditions on the complex values z1 and z2 does the equation hold for all complex values of w? Homework Equations The Attempt at a Solution...

Homework Statement Hi, I want to calculate the relative error between two complex numbers. Let's say z'=a' + i b' is an approximation of z = a + i b. Homework Equations How can I calculate the relative error between two complex numbers z' and z? The Attempt at a Solution...

Sketch on an Argand diagram the set of points satisfying both |z-4i|<=\sqrt{5} and \frac{\pi}{4}<=arg(z+4)<=\frac{\pi}{2}. I have already sketched the 2 loci. The problem lies in the following part. Hence find the least value of |z-2\sqrt{2}-4i|. Find, in exact form, the complex number z_1...

w is a fixed complex number and \( 0<arg(w)<\frac{\pi}{2} \). Mark A and B, the points representing w and iw, on the Argand dagram. P represents the variable complex number z. Sketch on the same diagram, the locus of P in each of the following cases: (i) \( |z-w|=|z-iw| \) (ii)...

Patterns found in complex numbers URGENT! use de moivre's theorem to obtain solutions to z^n = i for n=3, 4, 5 generalise and prove your results for z^n = 1+bi, where |a+bi|=1 what happens when |a+bi|≠1? Relevant equations[/b] r = √a^2 + b^2 z^n = r^n cis (nθ) This is what i...

In my math class, were having presentations about any topic from our curriculum. I want to talk about Complex numbers role in physics, but i don't know anything about its role. Can anyone tell me some areas were its important/used;) I know Feynman used them, but i don't know why. Anybody know...

Most of advance/modern physics has i(imaginary components like E and P are represented so ) in it..How does these imaginary co-ordinates or axes fit into application of physics which explains real world phenomenon..Hope my question sounds reasonable.?.THANK YOU IN ADAVANCE

The first part of the question asked to find the roots of w^5=1 which I have found to be e^{2k\pi)i} Hence show that the roots of the equation z^5-(z-i)^5=0, z not equal i, are \frac{1}{2}(cot{\frac({k\pi}{5})+i), where k=-2, -1, 0, 1, 2.

Given \(z=r(cos\theta+isin\theta)\), solve for \(z\) in the form \(a+ib\) if \(4+i(4z+1)=2re^{i(\pi+\theta)}\)

The complex number w has modulus \(\sqrt{2}\) and argument \(-\frac{3\pi}{4}\), and the complex number \(z\) has modulus \(2\) and argument \(-\frac{\pi}{3}\). Find the modulus and argument of \(wz\), giving each answer exactly. By first expressing w and \(z\) is the form \(x+iy\), find the...