4
votes
3answers
112 views

all complex solutions of $z\sin(z)=1$?

A possibly easy question, Can we find all complex solutions of $z\sin(z)=1$ ? My try: Let $$\sin(z) = \frac{e^{iz} - e^{-iz}}{2i}$$ so we have $$ z\frac{e^{iz} - e^{-iz}}{2i}=1 $$ Not sure how ...
1
vote
1answer
45 views

$\lim\limits_{R \rightarrow \infty}{|\cos z|}$ in complex numbers

Let $z=Re^{i \alpha}$ $\alpha$ is given number and $\alpha \in (0, \frac{\pi}{2})$ Find (if exist) $\lim\limits_{R \rightarrow \infty}{|\cos z|}$ What happen if function $f(z)=|\cos z|$ when ...
0
votes
2answers
45 views

Solving polynomial equation using known trigonometric identity

I recently did an exam paper in which the following question was asked: Prove $$\sin(5\theta)=16\sin^{5}(\theta)-20\sin^{3}(\theta)+5\sin(\theta)$$ Hence find all the solutions to: ...
3
votes
1answer
44 views

Why does the imag. part of the graph of $\zeta(n^{ix})$ resemble the tangent function?

If you input $\zeta(n^{ix})$ into the Wolfram Alpha search bar, in the plot, you get an infinitely repeating sinusoidal curve, which resembles the real part, and you get an infinitely repeating ...
1
vote
4answers
720 views

Lagrange's Trigonometric Identity

Lagrange's Trig identity is $$ 1+\cos\theta+\cos 2\theta +\cdots + \cos n \theta=\frac{1}{2}+\frac{\sin\frac{(2n+1)\theta}{2}}{2\sin \frac{\theta}{2}},\quad (0<\theta <2\pi). $$ How can we prove ...
2
votes
1answer
60 views

Is it valid to write $\sin(x+iy)=\sin (x)\cos(iy)+\cos(x)\sin(iy)$

I'm asked to sketch the set $\{z \in \mathbb{C}:\sin z$ is a real number $\}$ Here's what I did: $$ \sin(x + iy) = \sin(x) \cos(iy) + \cos(x) \sin(iy) $$ But since we only want the real part, then ...
2
votes
1answer
34 views

Show that the exists an unbounded open subset of the complex plane where sinz is bounded

$$ \sin z = \frac{e^{iz}-e^{-iz}}{2i} $$ So I think $ \sin z \le 1/2 + 1/2 = 1 $. But I am not sure how to find an unbounded subset of the complex plane, is it not bounded on the whole
1
vote
1answer
54 views

The function $e^{iz}$ maps $[0,2\pi)$ bijectively to $\{z\in\Bbb C:|z|=1\}$

Does anyone know a good proof or reference for this fact? I'm trying to write an elementary formal proof of this from just the series for $e^z=\sum_{n=0}^\infty z^n/n!$ (I don't have calculus ...
1
vote
1answer
22 views

Complex argument and nyquist plot

I'm trying to sketch the nyquist plot of $$\frac{j\omega-1}{j\omega+1}$$ but can't seem to calculate the argument correctly. I think it should be $$\arctan(-\omega) - \arctan(\omega) = ...
0
votes
2answers
44 views

Expressing the sine function in terms of exponential

Prove $e^{iz} - e^{-iz} = \sin z$. I used $$\begin{align*} \sin z & = z - z^3/3! + z^5/5! - z^7/7! + \dots & (i) \\ e^{iz} & = 1 - z^2/ 2! - iz^3/3! + \dots & (ii) \\ e^{-iz} ...
0
votes
0answers
20 views

What is the domain of complex tangent function?

What is the domain of $\tan z =\frac{\sin z}{\cos z}$ ? Is the domain $D=\{z\in\mathbb{C} : \cos z \neq 0\}$? Or considering the Riemann sphere, is $\tan z$ defined on $D$ as $\infty$?
1
vote
0answers
21 views

Trigonometry integration with a bound

So, I want to integrate $\int_\gamma sinz\; dz$ where $\gamma$ is any curve joining $i\to \pi$. Can I say that it is beacause $\int sinz=-cosz$, and $-cosz$ is analytic on the domain containing ...
3
votes
1answer
76 views

Evaluating trigonometric integral and Cauchy's Theorem

I am trying to evaluate the following integral: $\int_0 ^\pi {d\theta\over{1+\sin^2\theta}}$ I tried using the substitution of $\sin\theta={1\over 2i}(z-1/z)$, where $z=e^{i\theta}$, and ...
1
vote
1answer
28 views

Trigonometric functions (complex)

I have to find $sen^3{5a}$ and $cos^2{5a}$ considering that $sen{a}=\displaystyle\frac{1}{2}$ and $a$ belong to the first positive quadrant. I tried to apply De Moivre formula to find ...
0
votes
0answers
45 views

integral 2D involving complex exponential and cosine

I've some doubts about my solution of this integral: $$I(\phi_{1},\phi_{2})=\int_0^ {2\pi} \,d\phi_{1}\int_0^ {2\pi} \,d\phi_{2} \frac{e^{-in\phi_{1}} e^{-im\phi_{2}}}{2\pi}\frac{e^{il\phi_{1}} ...
0
votes
4answers
140 views

Show that $\cot(5\theta)=\frac{1-10\tan^2(\theta)+5\tan^4(\theta)}{1-10\tan^3(\theta)+5\tan(\theta)}, \forall\theta\in R $

Show that $$ \cot(5\theta)=\frac{1-10\tan^2(\theta)+5\tan^4(\theta)}{1-10\tan^3(\theta)+5\tan(\theta)}, \forall\theta\in R $$ using De Moivre's theorem.
4
votes
1answer
123 views

Convergence of $\sum\frac{\tan(nz)}{n^2}$ to an analytic function…what if $z\in \mathbb{R}$?

For which values of $z$ does $$\sum_{n=1}^\infty \frac{\tan(nz)}{n^2}$$ converge? For which values of $z$ is the limiting function analytic? One can show, as in this answer, that ...
0
votes
1answer
46 views

Condition on a sequence to generate an entire function

That conditions are necessary and sufficient for the succession $\{ \beta_n\}$ for the infinite product $$\prod_{n=1}^{\infty} \frac{ \sin(\beta_n z) }{\beta_n z} $$ converges an entire function. ...
2
votes
2answers
87 views

Find the argument of $ \frac{-1 + \sqrt3 i}{2+2i} $

I rewrite equation $ \frac{-1 + \sqrt3 i}{2+2i} $ as $$ \frac{ \sqrt3 - 1}{4} + \frac{ \sqrt3 + 1}{4} i $$ using the conjugacy technique. And set forward to find the argument of this complex ...
3
votes
0answers
169 views

Partial fraction development of $\cot \pi z$

"Compute the values $\sum_{n=1}^\infty \dfrac{1}{n^2}$ and $\sum_{n=1}^\infty \dfrac{1}{n^4}$ by comparison to the partial fraction development of $\cot \pi z$." I'm not sure what "the partial ...
0
votes
1answer
123 views

Why tan(1/z) has a non-isolated singularity at z=0?

Can someone please explain me this concept. Any sort of help will be highly appreciated.
0
votes
1answer
68 views

Representing a function as a real part of a complex variable?

To represent a simple sinusoidally varying function $V(t)$ let us use $V(t)=Re(\hat V e^{i\omega t})$ where $\hat V$ can be a complex constant.Let $\hat V =-iV_o$ Therefore, $V(t)=V_o \sin(\omega t)$. ...
0
votes
3answers
356 views

Find all complex solutions of $\sin(z)=1$ [closed]

Find all complex solutions of $\sin(z)=1$. How would I go about this?
2
votes
1answer
60 views

Sum of sines $\sum_{k=0}^{n} \sin(\phi +k\alpha)$

I've got the following problem. I'd like to prove that $$\sum_{k=0}^{n} \sin(\phi +k\alpha) = \frac{\sin\left(\frac{n+1}{2}\right)\alpha + \sin\left(\phi + ...
0
votes
1answer
38 views

Showing a certain series equals a certain trig function in complex analysis

I was reading a a complex analysis proof that was showing that for a fixed $\alpha$ s.t. $Im(\alpha)>0$ we have $\sum \limits_{m=-\infty}^{\infty} \frac{1}{(\alpha +n)^2}=(-4\pi^2)\sum ...
3
votes
3answers
483 views

Is it true that $ |\sin^2z+\cos^2z|=1, \forall z \in\Bbb C$?

We know that equation $ \sin^2z+ \cos^2z=1$ which holds $ \forall z \in\Bbb R$, actually holds $ \forall z \in\Bbb C$. Is it true that $ |\sin^2z+\cos^2z|=1, \forall z \in\Bbb C$? Thanks in ...
0
votes
0answers
90 views

How to Find End Point, after rotation

I am having an 3D object, length of the object is 27.5 meter, rotation value is -30 degree and the rotate origin point will be one end. After rotating the object i want to find the coordinate of ...
4
votes
1answer
112 views

An inequality involving arctan of complex argument

I have the following conjecture: \begin{equation} \text{Re}\left[(1+\text{i}y)\arctan\left(\frac{t}{1+\text{i}y}\right)\right] \ge \arctan(t), \qquad \forall y,t\ge0. \end{equation} Which seems to be ...
4
votes
1answer
222 views

Integrating $\int_0^{\pi/2} \cos^a(x) \cos(bx) \ dx$

Please help me in this integral : $$\int_0^{\pi/2} \cos^a(x) \cos(bx) \ dx \quad \text{if}\; b>a>-1$$ Please help me I used everything and can't evaluate it.
0
votes
1answer
68 views

Integrating trig functions with $R(\frac {z+1/z} {2}, \frac {z - 1/z} {2i} )$

Someone told me that there is a method for integrating rational functions $R(\cos{\theta}, \sin { \theta})$ by doing contour integration of the complex function $$\frac {R \left( \frac {z + \frac1z} ...
17
votes
5answers
1k views

Prove that $\sum\limits_{k=0}^{n-1}\dfrac{1}{\cos^2\frac{\pi k}{n}}=n^2$ for odd $n$

In old popular science magazine for school students I've seen problem Prove that $\quad $ $\dfrac{1}{\cos^2 20^\circ} + \dfrac{1}{\cos^2 40^\circ} + \dfrac{1}{\cos^2 60^\circ} + ...
2
votes
3answers
1k views

Express $\cos 6\theta $ in terms of $\cos \theta$

I think I'm supposed to use the chebyshev polynomials, as in $$ \cos n \theta = T_n(x) = \cos(n \arccos x)$$ But no idea what now?
7
votes
2answers
204 views

Interpolation using trigonometric polynomials of bounded modulus

Consider a grid of points $T=\{t_1,\ldots,t_m\}$ with $0\le t_i\le 1$. I would like to derive conditions on $t_1,\ldots,t_m$ (interpolation points) under which for any sequence of complex numbers ...
2
votes
1answer
147 views

How to compute $\int_0^\infty \frac{\sin t}{t^{s+1}} dt $?

How to compute $\displaystyle\int_0^\infty \frac{\sin t}{t^{s+1}}\;\text dt$ ? Here, the real part of the complex number $s$ is negative and greater than $-1$.
2
votes
1answer
87 views

Why is $e^{g(x)} = \pi$ where $g(x)$ is holomorphic in Weierstrass factorization of sine function?

Why is $e^{g(x)} = \pi$ where $g(x)$ is holomorphic in Weierstrass factorization of sine function? I just can't get why it's true.
2
votes
3answers
151 views

Incoherence using Euler's formula

Using the relation $\ e^{ix} = \cos(x) + i\sin(x)$ and substituting for $\ x = \pi$, we have the well-known Euler identity, $ e^{i\pi} = -1$. Substitute also for $ x = -\pi $, we have $ e^{-i\pi} = ...
2
votes
0answers
69 views

Bernoulli generating function and cotangent

May I ask for a little help in solving a problem about Bernoulli number generating function? Bernoulli number generating function is given by: $$f(z):=\begin{cases} \frac{z}{e^{z}-1} & z \in ...
3
votes
1answer
54 views

Question about an identity

Is it true that: $$ \coth^{-1}(z) = \tanh^{-1}\left(\frac{1}{z}\right), z\in \mathbb{C} $$ I used this identity: $$ \coth{z} = \dfrac{-1}{\tanh{z}} $$ To obtain such a result.
4
votes
1answer
283 views

how to prove $\displaystyle \frac{\sin (2n+1)\theta}{\sin \theta} = … $

How to prove $$ \displaystyle \frac{\sin (2n+1)\theta}{\sin \theta} = (2n+1) \prod_{k=1}^{n}\left(1 - \frac{\sin^2 \theta}{\sin^2 \left( \frac{k\pi }{2n+1} \right ) } \right ) $$ So far, I manage to ...
4
votes
2answers
166 views

proving $\csc^2 \left( \frac{\pi}{7}\right)+\csc^2 \left( \frac{2\pi}{7}\right)+\csc^2 \left( \frac{4\pi}{7}\right)=8$

How can I prove the following identity using complex variables $$ \begin{align*} 1) & \csc^2 \left( \frac{\pi}{7}\right)+\csc^2 \left( \frac{2\pi}{7}\right)+\csc^2 \left( \frac{4\pi}{7}\right)=8 ...
1
vote
1answer
216 views

Questions regarding finite Blaschke product

This problem is totally out of my ability. Not even sure what it is talking about. Somebody please help me to solve this... A finite Blaschke product of degree $n-1$ is a function of the form $ \ ...
6
votes
1answer
96 views

Can we give a definition of the cotangent based on a functional equation?

I've recently learned that the cotangent satisfies the following functional equation: $$\dfrac1{f(z)}=f(z)-2f(2z)$$ (true for $f(z)\neq 0$). Can we solve this equation for real or complex ...
11
votes
1answer
551 views

Can the real and imaginary parts of $\dfrac{\sin z}z$ be simplified?

I have calculated the real and imaginary parts of $\dfrac{\sin z}z.$ I've obtained $$\begin{eqnarray} \frac{\sin z}z&=&\frac{\sin(x+iy)}{(x+iy)}\\ &=& ...
0
votes
1answer
318 views

Evaluate$ \ \int_0^{2 \pi} \frac{\sin^2 \theta}{5 + 4 \cos \theta}\,d \theta \ $ using contour integration and the calculus of residues

Evaluate$ \ \int_0^{2 \pi} \frac{\sin^2 \theta}{5 + 4 \cos \theta}\,d \theta \ $ using contour integration and the calculus of residues
1
vote
1answer
116 views

Evaluate $ \ \int_{- \infty}^{\infty} \frac{x \sin (3x)}{x^2 +4}\,dx \ $ using Jordan's Lemma

How to to evaluate $ \ \int_{- \infty}^{\infty} \frac{x \sin (3x)}{x^2 +4}\,dx \ $ using Jordan's Lemma?
1
vote
1answer
158 views

Find all singularities of$ \ \frac{\cos z - \cos (2z)}{z^4} \ $

How do I find all singularities of$ \ \frac{\cos z - \cos (2z)}{z^4} \ $ It seems like there is only one (z = 0)? How do I decide if it is isolated or nonisolated? And if it is isolated, how do I ...
0
votes
1answer
153 views

Evaluate $ \ \int_{- \infty}^{\infty} \frac{x}{( x^2 + 4x + 13 )^2}\,dx . \ $ using contour integration and the calculus of residues

How do I evaluate $ \ \int_{- \infty}^{\infty} \frac{x}{( x^2 + 4x + 13 )^2}\,dx . \ $ using contour integration and the calculus of residues? I have many more problems of this kind need to be done. ...
1
vote
2answers
151 views

Use Residue Theorem to evaluate $ \ \oint_{C_3 (0)} \frac{z+7}{z^4 + z^3 - 2 z^2}\,dz \ $?

How do I use Residue Theorem to evaluate $ \ \oint_{C_3 (0)} \frac{z+7}{z^4 + z^3 - 2 z^2}\,dz \ $ where $C_3(0)$ is the circle of radius 3 centered at the origin, oriented in the counter- clockwise ...
2
votes
1answer
292 views

Singularities of $ \ \frac{z-1}{z^2 \sin z} \ $

Find all singularities of $ \ \frac{z-1}{z^2 \sin z} \ $ Determine if they are isolated or nonisolated. This is not hard, it is z = 0 and z = k*pi. But how do I: For isolated singularities, ...
2
votes
0answers
53 views

Rationality of Polynomial Coefficients. Integral Question.

We are entertaining polynomials with roots, all unique, on the curve $\Upsilon_s = \{{( 1-\cos[\theta])^{-s} \exp(i \theta)} \ | \ \theta \in \mathbb{R} \}$, where $s>0.$ $\Upsilon_s$ looks like a ...