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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4
votes
1answer
43 views

Find all complex numbers so that $Im(\frac{z+2}{2-i})=1$ and $Re(z^2+1)=1$ and for $z$ which is in the first quadrant find $\sqrt{z}$

Find all complex numbers so that $Im(\frac{z+2}{2-i})=1$ and $Re(z^2+1)=1$ and for $z$ which is in the first quadrant find $\sqrt{z}$. If $z=x+iy$ then $$\frac{z+2}{2-i}=\frac{x+2+iy}{2-i}\times ...
2
votes
1answer
43 views

Why is '1' the multiplicative identity of complex numbers and quaternions?

I am not a mathematician. I studied electrical engineering. I encountered quaternions while trying to understand motion of mobile robots and how rotations are achieved. This question occurred to me ...
6
votes
5answers
106 views

$e^{i\theta}$ versus $\cos\theta+i\sin\theta$

I am teaching an basic university maths course, and have been thinking about the complex numbers part. Specifically, I was wondering why I should include Euler's formula in my course. This led me to ...
1
vote
1answer
19 views

Separate real and imaginary part of $j \cos (z)$

Given the following expression $$w = j \cos \left[ \displaystyle \frac{1}{n} \arccos \left( \frac{j}{\epsilon} \right) + \frac{m \pi}{n} \right] = j \cos (z)$$ (which is related to this question; ...
1
vote
1answer
84 views

Question regarding complex numbers and real numbers?

I have two questions... If we take $(-1/3)^{(-1/3)}$ it would equal $-1.44224957$ since... $$(-1/3)^{-1/3}$$ $$\frac{1}{(-1/3)^{(1/3)}}$$ $$\frac{1}{-0.6933612744}$$ $$-1.44224957\ldots$$ Yet when I ...
1
vote
4answers
73 views

Find all complex numbers $z=a+bi$ such that $z^3=8$.

Find all complex numbers $z=a+bi$ such that $z^3=8$. I'll be happy if someone say me with what steps I have to start solving this problem.
0
votes
8answers
86 views

Trigonometric Property

How can I show that the following property holds? $2\cos(4a)+2\cos(2a)+1=\displaystyle\frac{\sin(5a)}{\sin(a)}$ I've been trying to derive it to no avail. What would be a way to approach similar ...
2
votes
1answer
34 views

Representation of Heaviside function's Fourier transform

I've seen here that the Fourier transform of Heaviside function $\Theta(t)$ is $$ \Theta(\omega) = \frac{1}{i\omega} + \pi \delta(\omega) \tag{1}$$ But in some physics texts and here I've seen the ...
2
votes
2answers
23 views

Property of polynomials proof

Let$$P(z)=\sum_{k=0}^n a_kz^k=a_0+a_1z+...+a_nz^n$$ be an N-th degree polynomial of a complex variable z, where the $a_k$ are complex constants. Now,$$\vert a_0\vert-\vert a_1\vert x-...-\vert ...
2
votes
0answers
31 views

Moving limit inside a contour integral

I'm trying to compute this integral as part of a larger problem I'm working on. I'm trying to solve the integral $\int_0^\infty \frac{\sin(x)}{x}dx$ and to do it I'm using the method where you ...
2
votes
2answers
43 views

Defining set of interior points of a triangle

Is there a way, given that $z_1,z_2 \ \text{and} \ z_3$ are the vertices of a triangle in the complex plane, to characterize all point that are inside of the triangle?
1
vote
3answers
77 views

Where's the mistake in this calculation? [duplicate]

Obviously something is wrong with this, but where is the error and why is it one? $$ \begin{align} \sqrt{-1} &= (-1)^{1/2} \\ &= (-1)^{2/4} \\ &= \sqrt[4]{(-1)^2} \\ &= \sqrt[4]{1} \\ ...
7
votes
2answers
134 views

proof for $\frac{1}{i} = -i$?

My physical chemistry textbook seems to be making the implicit assumption that $\cfrac{1}{i} = -i$. I'm not sure how this is valid. Here is the snippet of relevant steps: ...
7
votes
5answers
337 views

Definite integral of even powers of Cosine.

I'm looking for a step-by-step solution to the following integral, in terms of n$$\int_0^{\frac{\pi}{2}} \cos^{2n}(x) \ {dx}$$I actually KNOW that the solution is$${\frac{\pi}{2}} \prod_{k=1}^n ...
0
votes
0answers
18 views

Moving the absolute value inside of an integral involving a complex function

I have the following integral to evaluate $\lvert \int_0^\frac{\pi}{4}e^{iR^2e^{i2\theta}}iRe^{i\theta}d\theta\rvert$ and I want to put the absolute value sign inside of the integral so that I can ...
-2
votes
0answers
28 views

get real and imaginary part of incomplete elliptic integral

I am trying to evaluate the integral: $$\int \frac{1}{\sqrt{-z^2+k_1z^3+k_2z^4+k_3}} \;\mathrm{d}z$$ where $k_1=0.133, k_2=0.15, k_3=2.746$. The answer has been evaluated in MAPLE to 10 decimal ...
1
vote
4answers
29 views

Product of roots of unity using e^xi

Find the product of the $n\ n^{th}$ roots of 1 in terms of n. The answer is $(-1)^{n+1}$ but why? Prove using e^xi notation please!
2
votes
1answer
18 views

Finding the residue, $z=n\pi$, and $e^{n\pi}$

I have reached the following point in a residue calculation and am now unsure what to do: $$Res_{z= n\pi}=\lim_{z\to n\pi}\{(z-n\pi)\frac{ e^z}{\sin(z) } \}$$ $$=\lim_{z\to ...
8
votes
1answer
106 views

Cosh and Sinh analogs

We know that $$\cosh{x}+\sinh{x}=e^x$$ and that his can be expressed as $$\frac{e^x+e^{-x}}{2}+\frac{e^x-e^{-x}}{2}=\frac{(e^x+e^x)+(e^{-x}-e^{-x})}{2}=e^x$$ and this works out nicely because the ...
2
votes
3answers
55 views

Complex integration with trigonometric and logarithm

Show that $\int_0^{2\pi}\log\sin^22\theta dx=4\int_0^\pi\log\sin \theta d\theta=-4\pi \log2$ I did $$\int_0^{2\pi}\log\sin^22\theta d\theta=4\int_0^{\frac{\pi}{4}}\log\sin^22\theta d\theta$$ ...
2
votes
1answer
51 views

Complex sum using Laurent series?

By considering $f(z)=exp(z-\frac{1}{z})$ show that $$ \frac{1}{2\pi}\int_{0}^{2\pi}cos(n\theta-2sin\theta)d\theta=\sum_0^{\infty}\frac{(-1)^k}{k!(n+k)!}\ \forall n\ge1$$ f is holomorphic in ...
3
votes
1answer
37 views

Is the following function a constant function

Suppose that $f: \mathbb{C} \rightarrow \mathbb{C}$ is entire and bounded on the set $\{z \in \mathbb{C}; Re(z) \leq 0\}$. Is $f$ a constant function. I know by Picards theorem that a non-constant ...
6
votes
3answers
86 views

Entire function with uncountably many zeros

Suppose that an entire function $f$ has uncountably many zeros. Is it true that $f=0$? I have no idea how to proceed with this. Perhaps some theorem that I am not aware of. I have done an ...
4
votes
2answers
163 views

Complex Integration with trignometric function

Verify that $\int_0^{\frac{\pi}{2}}\frac{d\theta}{a+\sin^2\theta}=\frac{\pi}{2[(a(a+1)]^\frac{1}{2}}$ I know that $\sin\theta=\frac{e^{i\theta}-e^{-i\theta}}{2}$ then I did ...
0
votes
1answer
24 views

About the inequality $\frac{1}{2}>|\frac{z}{c}|, \forall z\in K.$

Let $c\geq 2 diam(K)$, where $K$ is compact in $\mathbb C$. Show that $\frac{1}{2}>|\frac{z}{c}|, \forall z\in K.$
0
votes
2answers
100 views

Trigonometric identities — a parallel RLC circuit connected to an AC-supply [closed]

An RLC-circuit is connected to an AC-supply as in the figure below. $I_{tot}(t)=I_0sin(\omega t+\phi)$ (denoted as $I_{ges} ( t)$ in the picture), $\phi$ is the phase angle between ...
1
vote
1answer
25 views

Showing there exists a complex differentiable function $g$ satisfying $g(z_0)=z_0$, with $g'(z_0) \neq 0$ and that $h(g(z))=(z−z_0)^{−m}$.

This is a follow up to a previous question: (Supposing $h$ has a pole, order m, at $z_0$, show the existence of a neighbourhood of $z_0$ and a new complex differentiable function $g$.) I'm trying to ...
1
vote
3answers
33 views

If real numbers $x$ and $y$ satisfy the equation $\frac {2x+i}{y+i}= \frac {1+i\sin{\alpha}}{1-i\sin{3\alpha}}$ then quotient $\frac xy$ is equal to?

If real numbers $x$ and $y$ satisfy the equation $\frac {2x+i}{y+i}= \frac {1+i\sin{\alpha}}{1-i\sin{3\alpha}}$, then quotient $\frac xy$ is equal to? Other conditions are ($\alpha \neq k\pi,\ ...
1
vote
1answer
44 views

why is the integer power of a complex number not multi-valued too?

my textbook [H. A. Priestley - Introduction to Complex Analysis] states about the argument of a complex number raised to a power : 'Only when $\alpha$ is an integer does $[z^{\alpha}]$not produce ...
2
votes
2answers
40 views

Improper integral and residues

Evaluate $\int_0^\infty \frac{dx}{x^4+1}$ By the residue theorem $$\int_{-R}^Rf(x)dx+\int_{C_R}dz=2\pi i\sum Res(f,z_i)$$ but I have problems to evaluate it because $$z^4+1=0\Rightarrow ...
1
vote
2answers
49 views

What does 𝔍(z) mean?

In complex analysis, if z is complex number, what does 𝔍(z) mean? The symbol is a mathematical fraktur capital J, unicode U+1D50D.
1
vote
3answers
18 views

Shortcut Technique for finding Raised Binomials with Imaginary Numbers

Find the Value of $(1+i)^5$ where $i$ is an imaginary number. The answer is $-4\cdot (1+i)$ We can always multiply them manually; but $i$ was wondering if there are any math tricks to quickly ...
-1
votes
0answers
52 views

Why doesn't -1 = 1 (spot the falacy) [duplicate]

I'm trying to figure out the fallacy in this statement: j = $ \sqrt{-1} = \sqrt{1/-1} = \frac{ \sqrt{1}} {\sqrt{-1}} = \frac 1j $ $ =>j^2=1 $ but $j^2 = -1 $ $ =>-1=1 $ My text book says ...
0
votes
1answer
30 views

Evaluate the improper integral with residues

Evaluate $\displaystyle\int_0^\infty\frac{dx}{x^2+1}$ I have that $z_0=i$ and $z_1=-i$ are singularity points but just $z_0=i$ is in the upper plane then ...
0
votes
2answers
44 views

Finding the complex roots of an equation.

I feel ridiculous asking this, its something I should be able to do, however I shall ask anyway. I am doing a calculation that requires me to find the roots of the equation ...
1
vote
1answer
18 views

Improper integrals and residues

I'm already read Conway, Churchill and Marsden but I'm still with doubts when it comes to improper integrals. Where come from this relation ...
0
votes
0answers
41 views

Primitive root of unity in complex plane

I have a polynomial $p(x) = -3x^{6}+ 4x^{5}-x^{4}-3x^{2} +6x-1$ in a complex plane and I need to transform it with DFT. Based on the degree of the polynomial makes ...
2
votes
1answer
46 views

Question on construction of entire functions

Suppose that $x_i$ and $y_i$ are sequences in $\mathbb{C}$. Can you construct a non constant entire function such that $f(x_i)=y_i$? What happens if $x_i$ have an accumulation point? or what happens ...
3
votes
1answer
38 views

Solving characteristic equation to find eigenvalue.

I came across the following question: The characteristic polynomial of a $3 \times 3$ matrix $A$ is $|\lambda I -A| = \lambda^3 + 3 \lambda^2+4 \lambda +3$. Find $trace(A)$ and $det(A)$. I know ...
2
votes
2answers
80 views
+50

Show that there is no analytic bijection from the unit disc to $\mathbb{C}$

Show that there is no analytic bijection from the unit disc to $\mathbb{C}$. I am quite unsure how to proceed here. I know for a fact that there is no analytic function from $\mathbb{C}$ to the open ...
-1
votes
1answer
53 views

What are hidden facts of Complex number? [duplicate]

I want to know how complex number can be used in real life. What are hidden usage of complex number in real life. Can anyone explain ? Thank you !
0
votes
0answers
16 views

Comparing the supremum of Maclaurian series with the function.

Suppose $f$ is an entire funciton with the Maclaurin Series $$a_0+a_1z+a_2z^2+\cdots $$ Show that if $r>0$ then $$|a_0|^2+|a_1|^2|r|^2+|a_2|^2|r|^4+|a_3|^2|r|^6+\cdots < \sup_{|z|=r} |f(z)|^2 ...
0
votes
1answer
19 views

Residues and poles show that

Show that i) $\displaystyle\operatorname{Res}\limits_{z=\pi i}\frac{z-\sinh z}{z^2\sinh z}=\frac{i}{\pi}$ ii) $\displaystyle\operatorname{Res}\limits_{z=\pi i}\frac{\exp(zt)}{\sinh z}+ ...
1
vote
2answers
44 views

Can someone explain in simple terms how to understand and calculate n to the power of i for n > 1? [closed]

I know how to add, subtract, multiply and divide any number by a complex number, but it is mysterious how one go about calculating $2$ to the power of $i$ for example. I would like to understand from ...
0
votes
1answer
36 views

Solving a complex number inequality involving absolute values.

Here is the relevant paragraph (from "Complex numbers from A to Z" by Titu Andreescu and Dorin Andrica) : Original question : How does $\left | 1+z \right |=t$ imply $\left | 1-z+z^2 \right ...
2
votes
1answer
43 views

question involving remainder of complex function

The question says - Dividing $f(z)$ by $(z-i)$, we get remainder $i$ and dividing by $z+i$, we get remainder $1+i$. Find the remainder upon division of $f(z)$ by $z^2 + 1$ How do I go about ...
1
vote
1answer
13 views

Studying electronic filters; how do I've to find the following complex argument limits?

$$\lim_{\omega\rightarrow0} \left(\arg\left(\frac{a+b+\left(i\omega l\right)+\left(\frac{1}{i\omega c}\right)}{a+b+f+g+\left(i\omega l\right)+\left(i\omega L\right)+\left(\frac{1}{i\omega ...
1
vote
2answers
27 views

Solving inequalities on both sides with complex numbers

I need to sketch this region $\left \{ z\in\mathbb{C}| |z-i|\leq |z-1| \right \}$. I'd like some assistance with solving this inequality because I think that's where I'm going wrong. To solve the ...
7
votes
2answers
73 views

Imaginary $\cos^{-1}$ value significance?

When I was bored in AP Psych last year, I jokingly asked myself if there was a cosine inverse of $2$. Curious about it, I tried calculating it as follows: $$ \begin{align*} \cos (x) &= 2 \\ \sin ...
5
votes
6answers
110 views

Proof of Euler's formula that doesn't use differentiation?

So I saw a 'proof' of the sine and cosine angle addition formulae, i.e. $\sin(x+y)=\sin x\cos y+\cos x \sin y$, using Euler's formula, $e^{ix}=\cos x+i\sin x$. By multiplying by $e^{iy}$, you can get ...