# Questions tagged [complex-numbers]

Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

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### Compute $f(z)$ and show it's well defined.

I'm given the function $f$ in integral form as follows $$f(z)=\int_{\gamma} \frac{dw}{w-z},$$ where $\gamma(t)=t, \, 0<t<1$ and $z \notin [0,1]$. I'm asked to compute this integral and show that ...
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I have to do such a question: Let ABC be a triangle, whose vertices A, B, C correspond to the complex numbers α, β, γ (the origin is not necessarily at one of the vertices), respectively. Let $ω = e^... 0 votes 2 answers 57 views ###$Z=\ln(i)/i$, find$Z$. It feels like a trivial question, but i don't know if my answer is correct. My attempt : $$Z = \frac{\ln(i)}{i}$$ $$iZ= \ln(i)$$ $$e^{iZ} = i$$ $$\cos Z + i\sin Z = i$$ $$\cos Z = i(1-\sin Z)$$ ... -2 votes 0 answers 25 views ### When does convergence of the difference of the moduli of two complex sequences imply the convergence of the difference of the two complex sequences Let$a_i$and$b_i$be convergent sequences of complex numbers such that (i) for all$i$,$|a_i|, |b_i| \leq 1$, (ii)$a_i \neq e^{i\theta}b_i$, (iii) the real sequence$c_i = |a_i| - |b_i|$converges ... 0 votes 0 answers 51 views ### is$\frac{(-1)^kk!}{(s+a)^k}$ODD? Is the following $$\frac{(-1)^kk!}{(s+a)^{k+1}}$$ an odd function?$f(-s) = -f(s)$? It looks like the function does$\mathbb R\to \mathbb R$, but not$i\mathbb R\to i\mathbb R$(pure imaginary), so ... -3 votes 1 answer 65 views ### prime factorization and complex number fields [closed] if we define complex number fields in the following manner: ( a + b( (-1)^(1/n) ) ) it would appear as if n = 2 is the only value for which unique prime factorization holds, atleast in reference to a^... -2 votes 1 answer 89 views ### Find all complex numbers$𝑧$for which$𝑧^6+5𝑧^3+6=0$Find all complex numbers$𝑧$for which$𝑧^6+5𝑧^3+6=0$this is a question we had in class but I'm somehow confused/lost in the explanation of how to solve it. I'd very kindly appreciate it if you ... -1 votes 1 answer 139 views ### How to evaluate:$\int_0^1(-1)^{\ln(x)}dx$So I was looking through the homepage of Youtube to see if there were any integrals that I might want to evaluate when I came across this video by Maths$505$which showed how to evaluate$$\int_0^1(-1)^... 0 votes 2 answers 53 views ### Translation/Rotation properties for equations in the complex plane Prove that complex numbers (z_1,z_2,z_3) that satisfy the relation below form an equilateral triangle in the complex plane.$$z_1^2 + z_2^2+z_3^2=z_1z_2+z_2z_3+z_3z_1$$This answer first shows that ... 0 votes 0 answers 23 views ### How to simplify the complex expression of the Fourier series for a square wave? I'm trying to find the expression of the fourier series (using complex numbers) for the following odd squarewave function with period T: V(t) = V_0 for 0<t<T/2 V(t) = -V_0 for T/2<t&... 0 votes 0 answers 33 views ### How do we prove this nested radical solution? I’m submitting this question because my previous one was perhaps too verbose and did not get the point very quickly. A well known \sqrt{2} nested radical has the following solutions as found in the ... 1 vote 2 answers 107 views ### Why can we plug complex numbers into maclaurin series? When finding a Maclaurin series for a function f(x). We evaluate f '(0), f ''(0), etc, to find our coefficients for each term. I have done this for the standard functions, f (\mathbb{R}):->\... 2 votes 1 answer 65 views ### n complex numbers inside a disk with center A and radius 1 Inside the disk with center A(2,0) and a radius of 1, a set of n \geq 1 points is considered, each having the respective affixes z_1, z_2, \ldots, z_n. Show that \left| \sum_{i=1}^{n} z_i \... -1 votes 0 answers 34 views ### Solved a complex equation, but got 2 answers, which is the right one? The problem and my half solution z^4+2(i+1)z^2+i=0 So i did: x^2+2(i+1)x+i=0 \rightarrow x_{1,2}=\frac{-2i-2\pm\sqrt{8i-4i}}{2}=? \sqrt{4i} => \sqrt{4}\cdot(cos\frac{\pi/2+2k\pi}{2}+i\cdot sin\... 4 votes 1 answer 89 views ### Inequality with complex numbers with the same modulus I got stuck at the following problem. Prove that for every three distinct complex numbers a, b, c with |a| = |b| = |c| > 0, the following inequality holds: \sum_{\text{cyclic}} |(a+b)(b-c)... 4 votes 1 answer 196 views ### Inequality for a complex polynomial Let p(z), q(z), and r(z) be polynomials with complex coefficients in the complex plane. Suppose that |p(z)|+|q(z)| \leq|r(z)| for every z. Show that there exist two complex numbers a, b such ... 2 votes 1 answer 33 views ### if |z-3-2i| \leq 2 , \text{ find the minimum of } |2z-6+5i| if |z-3-2i| \leq 2 , \text{ find the minimum of } |2z-6+5i| My attempt:- Let z=x+iy so we have (x-3)^2+(y-2)^2 \leq4 let |2z-6+5i | \text{ be }\phi so {\phi}^2 =4(x-3)^2+(2y+5)^2 I ... -1 votes 1 answer 89 views ### find the region where \frac{i}{z-z^7} is analytic I have tried it by putting z=x+iy Then reduced the whole into u(x,y)+i\cdot v(x,y) form. However, the expression I am getting is quite complicated and would be hard to put put in the CR eqn and ... -3 votes 0 answers 36 views ### Determine the points where \frac{z}{z-1} is analytic. I have tried it by puting z =x+iy then reduced the whole fn to u(x,y)+iv(x,y) form. However, the expression I am getting is quite complicated and would be hard to put put in the CR eqn and Laplace ... -2 votes 1 answer 95 views ### Is any value of (\sqrt i)^{\ln(i)} real? Is any value of (\sqrt i)^{\ln(i)} real? Inspired by similar expressions, I was playing around a bit on julia prompt and typed (\sqrt i)^{\ln(i)} ... 0 votes 0 answers 16 views ### A Nested Radical Arising from a Nonlinear Recurrence I have been looking into analytic continuation of regular recurrence relations into the negative, real, and complex domains (\mathbb{N} \mapsto \mathbb{Z}, \mathbb{R},\mathbb{C}). In doing so, we’ve ... 0 votes 0 answers 40 views ### Solving for the derivative at the point z = 0 for the function f(z) = z^2 - \bar{z}^2 I'm trying to solve for the derivative of f(z) = z^2 - \bar{z}^2 using the limit definition (I'm positive that it can be done with power rule, but given my track record recently, I'm not going to ... 0 votes 1 answer 48 views ### How to solve complex equation with i in the bx term? First of all, sorry for the English, i have never asked question about math in English before, so i don't know if my problem is understandable. So, basically i don't know anything about this stuff, i ... 1 vote 0 answers 64 views ### Removing "i" from the triangle inequality? In my complex analysis class, I have to prove that a limit exists for a function and I think I can use the Triangle Inequality in my proof, but I don't know if it's possible. My question: knowing that ... 2 votes 1 answer 72 views ### Must every Cauchy-Riemann condition be fulfilled simultaneously? Working through problems in my complex analysis book, and I have to determine where the derivative exists for a function. I know that the derivative can exist only along a certain curve, however I don'... 1 vote 1 answer 47 views ### Finding the number of distinct common roots of unity Consider equations:$${x}^{p} = 1\\ {x}^{q} = 1$$Then the number of common roots is equal to the \gcd(p,q). But, how can I prove this statement? 4 votes 0 answers 88 views ### Confinement result for unit complex numbers Inspired by this problem, and some computer simulations, I almost convinced myself of the following result. However, I am coming short on a proof. Result: Let n\geq 3, and 2n+1 complex numbers z_{... -4 votes 0 answers 75 views ### Can someone teach a class 10th student calculus or advanced mathematics? Like starting from what are functions to the end? [closed] I like math, even though the level of math I do is rather easy but I like learning new things in math and envy people who solve complex equations on their copy or board or whatever they are using ... 1 vote 2 answers 83 views ### If z is a non-real complex number , find the minimum value of \frac{\operatorname{Im}(z^5)}{(\operatorname{Im}(z))^5} If z is a non-real complex number , find the minimum value of$$\frac{\operatorname{Im}(z^5)}{(\operatorname{Im}(z))^5}$$where \operatorname{Im}(z) is the imaginary part of a given complex number ... 0 votes 1 answer 63 views ### Is there a way of describing why multiplying complex numbers adds their angles intuitively? Everywhere I'd looked for an explanation of this angle-adding phenomenon, it seemed to have been in one of two forms: Either something roughly like this:$$\left(\cos\left(a\right)+i\sin\left(a\right)\... 0 votes 0 answers 24 views ### Critical point of complex function An exercise I'm doing is asking me to show$\partial_z f=\partial_{\bar{z}}f=0$is equivalent to$\partial_x f=\partial_{y}f=0$using these two definitions: $$\partial_{z}z^{n}=nz^{n-1}$$ and $$\... -2 votes 0 answers 41 views ### Roots problem in linear equations by using complex numbers Let there be an equation$$x = \sqrt{144}$$This can be written as$$x = \sqrt{(-12)^2}$$We know that,$$\sqrt{k^2} = \sqrt{k} \times \sqrt{k}x = \sqrt{-12} \times \sqrt{-12}x= \sqrt{12} \... -2 votes 0 answers 30 views ### i^2 is not equal to -1? [duplicate] as$i = \sqrt{-1}$so$i^2 = -1$however i seem to have thought of a counter argument $$1 = \sqrt1 = \sqrt{(-1)(-1)} = i^2 = -1$$ but 1 can't be equal to -1 I must be going wrong in the$\sqrt{(-1)(-1)...
Let $S$ be the unit circle in the complex plane, $$S = \{z \in \mathbb{C} : |z| = 1\}.$$ For values $z_1^{(1)},z_1^{(2)},z_2^{(1)},z_2^{(2)},\ldots,z_k^{(1)},z_k^{(2)} \in S$, letting \begin{align*}...