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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2
votes
1answer
20 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
114 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
52 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
39 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
90 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
170 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
25 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
112 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 $h(g(z))=(zāˆ’z_0)^{āˆ’m}$.

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
34 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
41 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 š”(z) mean? The symbol is a mathematical fraktur capital J, unicode U+1D50D.
1
vote
3answers
20 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 ...
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
19 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
43 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
39 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
84 views

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
55 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
37 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
14 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
74 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
115 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 ...
2
votes
2answers
23 views

Showing internal angles of a square are unaffected by a mapping

I recently had an exam in complex analysis, and I am slightly confused by one of the questions, so I'd appreciate any clarification: The mapping from the complex $z$ plane to the complex $w$ plane ...
1
vote
1answer
73 views

Why is the Euclidian norm used to measure complex numbers?

Why is the Euclidian norm used to measure complex numbers? The complex numbers are numbers (or more precisely, pairs of numbers), and I can't see why are they essentially connected to the ...
0
votes
2answers
61 views

Paradox - minus one equals one using square roots [duplicate]

I was looking on Howard Eves's book "An Introduction to the History of Mathematics" and I stumbled upon a demonstration on how $-1 = 1$. The demonstration follows: $$ \sqrt{-1} = \sqrt{-1} $$ $$ ...
1
vote
0answers
52 views

complex rank-one update

I'm trying to find the eigendecomposition of a rank-one update to a complex matrix $D + uv^T$. The matrix $D$ is diagonal, but not the identity. It has unique imaginary entries along the diagonal. ...
1
vote
1answer
37 views

Evaluating residua and simplifying complex expressions.

My question is in two parts, so please forgive its long-winded nature. Lets say that I want to find the residua of the following complex function: $$f(w)=\frac{2w+1}{w(w^3-5)}$$ Let us, ...
2
votes
1answer
56 views

Find the poles/residues of $f(z)=\frac{\sin(z)}{z^4}$

I'm trying to calculate the residua of the following complex function but am encountering problems trying to determine its poles: $$f(z)=\frac{\sin(z)}{z^4}$$ Expanding the denominator shows that we ...
1
vote
1answer
68 views

Evaluation of Residua

Suppose that I have the following complex valued function, and want to evaluate its residua: $$h(z)=\frac {z^5}{(z-3)(z^4+2)}$$ For both parts of the denominator we will have simple poles. For our ...
2
votes
2answers
84 views

How to prove that $\cos^2(z)+\sin^2(z)=1$, where $z$ is a complex variable (if it is true)?

Can someone show me: If $x$ is a real number, then $\cos^2(x)+\sin^2(x)= 1$. Is it true that $\cos^2(z)+\sin^2(z)=1$, where $z$ is a complex variable? Note :look [this ] in wolfram alpha ...
1
vote
1answer
30 views

prove that $\Bbb Z/n \Bbb Z \cong \mu_n$

I need to prove that $\Bbb Z/n \Bbb Z \cong \mu_n$ $\Bbb C^x \gt \mu_n = \{z \in \Bbb C^x | z^n = 1 \}$ what i tried - I tried building a homomorphism $f: \Bbb Z \to \mu_n$ such that $f(z) = e^{{2 ...
2
votes
3answers
60 views

Why $\lim_{R\to\infty}\int_{0}^{\pi}\sin(R^{2}e^{2i\theta})iRe^{i\theta}\:\mathrm{d}\theta = -\sqrt{\frac{\pi}{2}}$

This is a short question, but I'm simply not sure where to start, I know by Jordan's Lemma that the integral is not $0$, but I only know the below result due to Mathematica. ...
3
votes
1answer
65 views

If $a+b+c$, $a^2+b^2+c^2$, $a^3+b^3+c^3$ are real, then so are a,b,c

Let $a,b,c$ be complex numbers with distinct magnitudes such that $a+b+c$, $a^2+b^2+c^2$, $a^3+b^3+c^3$ are real. Prove that $a,b,c$ are real numbers as well. I tried to go for ...
1
vote
2answers
41 views

All solutions of $z\in \Bbb C, \cos z = i$

I want to find all solutions to $\cos z = i$ Okay so $$\cos z = \frac12(e^{-iz}+e^{iz})=i$$ $$e^{-iz}+e^{iz}=2i$$ $$e^{-iz}+e^{iz}=2e^{\frac\pi 2i + 2\pi i n},n\in \Bbb Z$$ $$1+e^{2iz} = ...
3
votes
3answers
122 views

Why is $\sqrt{-i} \neq i\sqrt{i}$?

I wanted to figure out the square root of $-i$. Since $\sqrt{-x} = i\sqrt{x}$, $\sqrt{-i}$ should equal $i\sqrt {i}$, however, WolframAlpha said it was false. However, if I do say that ...
3
votes
3answers
87 views

Is $|z+i| = |z-1|$ a circle with radius $\sqrt{1^2+1^2=1}$ and origin $(1,-i)$?

Is $|z+i| = |z-1|$ a circle with radius $\sqrt{1^2+1^2}=1$ and origin $(1,-i)$? Because I know $|z+i| = 3$ is is a circle with radius $3$ and origin $(0,i)$.
-4
votes
3answers
94 views

1 is equal to -i [duplicate]

Pretty simple, but I'm sure there's some subtlety to it I'm missing. $$(-i)^2=1 \Rightarrow \sqrt{1}=-i \Rightarrow 1=-i$$ Looking at an Argand diagram however, gives some reason to doubt this. ...
1
vote
1answer
42 views

Analytic function for which $\overline{f(z)} \neq f(\overline{z})$?

Since $\overline{f(z)} = f(\overline{z})$, where $\overline{z}$ denotes the complex conjugate of $z$, already works for polynomials with coefficients in $\mathbb{R}$, the exponential function, etc., ...
1
vote
3answers
74 views

System of equations with complex numbers-circles

The system of equations \begin{align*} |z - 2 - 2i| &= \sqrt{23}, \\ |z - 8 - 5i| &= \sqrt{38} \end{align*} has two solutions $z_1$ and $z_2$ in complex numbers. Find $(z_1 + z_2)/2$. So far ...
-2
votes
2answers
42 views

please help me to solve $\frac{1-\exp(-10j\pi)}{1-\exp(-2j\pi)}$

Why the result of the below statement is equal to $5$? $$\frac{1-\exp(-10j\pi)}{1-\exp(-2j\pi)}$$ I compute this way and it results NaN! $1-\exp(-j \cdot 10 \cdot \pi)= 1-(\cos(10 \cdot \pi)-j ...