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#cot(pi/2+theta)
truphysics · 1 year
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Problem 10.1 - Griffith's Intro to QM
Problem 10.1 Rutherford scattering. An incident particle of charge $q_1$ and kinetic energy $E$ scatters off a heavy stationary particle of charge $q_2.$ (a) Derive the formula relating the impact parameter to the scattering angle. Answer: $b=(q_1q_2/8\pi\varepsilon_0E)\cot(\theta/2)$ (b) Determine the differential scattering cross-section. Answer: $D(\theta)=\left[…
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devphilamaths · 3 years
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Solve cot(pi/2-theta) | cot(pi/2 -x) | cot pi/2 - x formula, Find value cot pi by 2 - x 
Hi friends.. In this tutorial Find the Value of cot(pi/2-theta) cot(pi/2 -x)cot pi/2 - x formula Find the value of cot pi by 2 -xTRIGONOMETRIC IDENTITIESExp...
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theladyofdeath · 7 years
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Castaway {ACOTAR/Chapter 2}
Word Count: 2,281
Summary:  A modern-day University AU, from the A Court of Thorns and Roses universe. All characters belong to Sarah J. Maas. The idea for this fanfic hailed from prompts sent in by Anonymous, and @queen-archeron. You can read previous chapters here.
Author’s Note: Chapter 3 is when shit starts getting real, so I hope you enjoy the path to feels. Let me know what you think. :)
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September 3 – The Day I Wore Heels
My paintbrushes look good in the vase Elain bought me. She probably meant for me to put flowers in them, but I guess that’s where we differ in taste.
My roommates are nice, I guess. Well, the blonde one is, anyway. Her name is Mor, and she changes her clothes at least fifteen times a day. Her make up is always immaculate, she never has a hair out of place, and I swear she wakes up that way. She keeps giving me strange looks, though.
They both do.
I’m not sure why.
I’ve racked my brain a thousand times of what I’ve done to offend them. I keep going back to Tamlin. They reacted strange when they found out we were dating. I didn’t ask, though. I can’t decide if I didn’t ask because I didn’t care to know why, or because I was too scared to know why.
What if he and Mor dated? I can’t see him dating someone like Amren.
Mother help the person who ends up with that one.
Maybe I’ll ask Tamlin tonight. He invited me to a party. I guess I finally get a chance to wear the heels Elain bought me for my birthday last year.
Once again, shoes are where we differ in taste.
 Love,
The Lamb in the Lion’s Den
 Feyre had never been to a college party before. Hell, she’d never been to a high school party, either, but that wasn’t the point. The point was, she was dressed in her only dress – a hip hugging, black, long-sleeved dress that reached only to her mid-thigh – and she was walking around campus, from her dormitory to the Beta Theta Pi fraternity house.
It didn’t take her long to find, considering music was blasting from its lawn and a banner with the frat’s name painted across it hung from its roof.
“Feyre!”
Feyre knew that voice. She turned to find him running toward her, his golden hair hanging limply in his eyes. “Hey.”
“You made it just in time,” he assured her.
“In time for what?” she glanced around, but it didn’t look like anything too exciting was about to take place.
“In time to see me before I have to make my rounds.”
Tamlin was the president of Beta Theta Pi, and he took his role very seriously. Feyre, to her disappointment, had not seen a lot of him since she moved to campus the day before.
“Let me show you around,” Tamlin smiled, and Feyre couldn’t say no. He took her hand, and escorted her through the door of Beta Theta Pi.
Feyre wasn’t sure what she was expecting a fraternity house to look like, but this wasn’t it. It looked plain, with posters hanging all over the walls, some of them a bit suggestive, and couches with holes torn in them – she didn’t care to know how said holes appeared.
And yet, it was clean.
There was a DJ in the corner, doing his thing, and smoking what smelled too much like a skunk’s ass to be a cigarette, as people began trickling through the door.
“There’s the kitchen, you are obviously in the living room, and, if you need it, down the hall to your left is the bathroom.” Feyre nodded, but her mind began racing as every new person entered the house. Her hand still in his, Tamlin led Feyre up the staircase. He didn’t say another word until they were enclosed in his bedroom.
Feyre blinked.
“What’s wrong?” he asked, after flipping on the light switch and sitting on the edge of his bed.
“Nothing!” Feyre said, but didn’t move from her spot, just inside the door. “I just – I don’t know, it’s plain in here.”
Tamlin looked around. It was plain. There was nothing on the walls, not too many belongings strung throughout, and a bland white comforter lounging on top of his mattress. “I think it looks fine.”
Feyre nodded, but didn’t say anymore.
“Come sit with me,” Tamlin offered, patting the mattress beside him, gently.
Feyre did.
“You seem nervous.”
Feyre shrugged. “This is new territory for me.”
Tamlin smiled, and placed a hand on her upper thigh. Feyre’s nerves calmed at his touch, as he kissed her cheek, softly. “I’m glad you’re here.”
It was short lived, though, because before Feyre could respond, Tamlin’s door was thrown open. “You’re needed.”
Tamlin gave his friend a short nod before rising to his feet. “Stay in here for a while. I’ll come get you when I’m done.”
Now, Feyre’s brows really lifted. “You want me to stay in here? Alone?”
“Trust me,” he assured her. “You don’t want to go out there and wander alone. I won’t be long. I promise.”
Before Feyre could agree, he left.
And closed the door behind him.
“How are you feeling?”
Nesta stood at the foot of his cot, as he watched her with a thirsty, lingering expression in his hazel eyes.
“Great,” he smiled, lazily. “As good as one can feel when they got hit by an intoxicated jackass who ran a red light.”
He’d given her the same answer every time she asked. She had to admit she was curious about what had happened, as he had always been very brief with his statements. But, it was not her business. You don’t care, she reminded herself. Continuously.
“Here.” she handed him his pain medication, and a glass of water, before jotting down his vitals. “Everything is looking good, Mr. Nazari. You should consider yourself very lucky.”
“I’ve told you to call me Cassian.” Oh. He had. Every five minutes.
Nesta simply nodded before taking his empty cup and retrieving his street clothes from the cabinet. There was blood stained on his jeans, Nesta noticed, although his tee-shirt seemed to make it through the accident unscathed. Beneath his neatly folded pile of clothes were bulky, black boots that looked as if they had been worn every day for years. Then, in a clear, plastic Ziplock bag were a set of keys, his wallet, and a dogtag necklace.
“Are you going to stare at my stuff, Nurse Nesta, or may I get dressed and out of this hideous blue nightgown you all have put me in?”
Nesta didn’t turn before she could feel that the scarlet that had brushed her cheeks had vanished. “Apologies, Mr. Nazari. I was only making sure I didn’t miss anything.”
“Cassian,” he corrected, then smiled. “Thank you. Would you mind helping me untie?”
It wasn’t until he lifted himself up with his arms, and scooted around so that he was sitting with his back to her, that Nesta realized what he was asking. Trying to keep her slender hands as steady as possible, Nesta Archeron began to untie her patient’s hospital gown.
The top one wasn’t bad, as it only revealed the top of his back, the broadness of his shoulders. The tie that was neatly in a bow in the middle of his back, though, let out a little more. The arms of the gown fell to his wrists, revealing a tanned chest, and an impressively toned back and arms that were covered in tattoos. She couldn’t stop her eyes from traveling down his back, then to her displeasure, she found he was wearing a pair of red boxer-briefs.
She cleared her throat. “I am sure you can get the last one yourself.”
A soft laugh shook his shoulders. “If you insist, Nurse Nesta.”
He shrugged his tee-shirt over his shoulders, not waiting for her to leave first. She took it as a hint to dismiss herself.
“Nurse Nesta?”
She froze in the doorway. “Yes, Mr. N-“
He stopped her by clearing his throat.
She sighed. “Cassian.”
When she turned, he looked pleased. “That wasn’t so hard, was it?”
He had no idea.
“I have a problem.”
Nesta attempted to ignore the fact that his gown was completely removed and discarded, and her patient was sitting in his underwear.
“Yes?” she asked, when she realized he was not going to continue until he had her full attention.
Cassian looked from his jeans, to the cast that reached from his toes to just below his knee. “How am I supposed to get my jeans over this cast?”
Nesta blinked. Four years of nursing school and sixty thousand dollars in debt……and they neglected to teach her that one.
A knock came from the door and a man, who looked eerily similar to Cassian, just with shorter hair, stepped into the room.
Nesta had seen him before, in pictures Elain had posted on the internet. Azriel.
“Az?” Cassian raised his brows. “Not that it isn’t a pleasure to see you, because it always is, but where is Rhys?”
“Car broke down at the University.” Azriel’s words were short, and quiet. “He’s our next stop.”
Then, something in him softened as he took in Cassian, broken and bruised and pant-less on the cot. “You okay?”
“Yeah,” Cassian said, and Nesta wondered what it was about this Azriel that could make him speak so seriously. “I’m fine, Az.”
Azriel nodded, then grabbed the crutches from the corner and handed them to Cassian. “Let’s go get you released, then.”
Cassian’s teasing grin returned as he rose to his foot, and swung his arms over the crutches, tossing his jeans over his shoulder as he did so.
Azriel didn’t even question his friend’s lack of pants.
“Well, Nurse Nesta, I guess this is how we say goodbye.”
Nesta gave him a nod. “Go to the end of the hall and turn left. You’ll see the checkout desk.”
Cassian tilted his head, and approached her, one hop at a time. “Unless you don’t want to say goodbye just yet.”
“Mr. Naza-“
“Cassian.”
“Cassian.” Nesta glanced at Azriel, who was looking between the two curiously. “I hope you feel better. Have a good night.”
Cassian grinned. “Is that all?”
Nesta nodded, crossly.
“Alright. Well, if you change your mind, I’ll be at the Harris Street Park on Thursday afternoon. By the pond. Probably sitting on a bench, considering.”
Nesta didn’t say anything else as the two walked out of the room, and down the hall.
She couldn’t believe that she watched him until he disappeared around the corner.
­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­Elain admired herself in the mirror.
She was excited. She should have been excited. She hadn’t been on a date in years.
Lucien had come back the night before, to The Fawn, and they had talked for hours. He’d asked if he could take her to dinner, and she would have been stupid to refuse.
There was guilt, though, and it was eating her alive.
Her and Azriel had a fight.
He had come by that morning, bringing her a blueberry muffin from the bakery across the street. When she told him that she had a date, his demeanor changed.
“With Lucien?” he had asked.
“Yes,” Elain replied, simply. “I don’t understand why you hate him so much.”
He opened his mouth, then closed it, then opened it again, as if he had changed what he was going to say. “I just don’t think he’s right for you is all.”
“No? And why is that?” Elain crossed her arms.
“He could be dangerous.”
“He could be dangerous?” she repeated, unimpressed. “And how did you come to that conclusion?”
“He hangs around the wrong crowd.”
“And you know that, how?” she threw her hands in the air, calmness thrown out the window. “What, like you know him so well.”
“Actually, I –“
“I think you should leave.”
Azriel froze, blinked, and raised his brows. “What?”
“You should leave.”
“You’re getting mad at me?” he asked, incredulously. “Elain, you know me way better than you know him –“
“You’re not acting like you right now. You’re acting like an idiot.”
Azriel simply nodded. “Fine. You know what? Do whatever you want. I have to pick Cassian up from the hospital in an hour. I should leave anyway.”
Elain didn’t say anything else, she simply pointed to the door.
Setting her muffin on the countertop, he left, shutting the front door with an eerie silence as he did so.
The muffin was still there.
Elain couldn’t touch it, couldn’t eat it, she felt so horrible about what had happened.
Azriel was out of line, though. He was acting foolish, and he knew it, men were just too stubborn to admit such faults.
Then again, Elain did know Azriel better than anyone, and he had never acted like that before. Maybe there was something he knew. Maybe she should have heard him out.
Elain shook her head, and observed her appearance, once more. She was wearing one of her own designs, a floral maxi dress of olive-green and blush, the perfect contrast to her dark brown braid.
She would have fun.
She would not think about Azriel, or their fight.
Lucien Vanserra was taking her out for a good time, good food and good company. She had the right to enjoy herself.
And yet, the guilty feeling lingered in the pit of her stomach, even as the doorbell rang.
Feyre was tired of waiting.
She’d been in his room for an hour, and there wasn’t much to do. The only personal things of Tamlin’s she was able to find were a few family pictures and magazines of old cars and medieval weaponry.
After the sun had finally set, she finally came to the conclusion that Tamlin was not returning any time soon.
So, she left.
People were dancing and grinding and hiding in corners with one another. Her boyfriend was nowhere to be found, not to her surprise.
She wasn’t sure she even wanted to witness Tamlin in such an environment.
“Excuse me. Coming through. Excuse me.” Feyre made her way through the crowd of dancers and oncoming intoxicated underaged drinkers until she was out of the door, in the front lawn.
If inside was crazy, outside was utter chaos.
Red solo cups were a necessity to fit in. The cloud of smoke Feyre walked through when finding her way around the corner had her in a coughing fit. And Tamlin –
Tamlin was in the center of it all.
The backyard was lit with string lights, and a crowd of people were taking advantage of the music in the air and alcohol in their systems.
Tamlin was in the middle of the grass, a girl grinding on his backside, and another on his front.
Nausea brewed in the pit of Feyre’s stomach as tears began to sting her eyes. Why had he asked her to come if he was only going to keep her away, while he was doing this?
If this is what his duties entailed, Feyre wanted nothing to do with it.
Although, if she wanted nothing to do with it, then why did it hurt so bad?
Feyre didn’t hesitate before turning on her heels, and heading toward the street. She removed her heels, so she could move quicker.
“Asshole,” she mumbled, under her breath, just before halting. “Shit.”
She’d left her purse.
In his bedroom.
Turning around, reluctantly, Feyre grasped her black high-heels as she trudged back toward the front door of the fraternity house.
“W-Who is this beautiful lady?”
Feyre froze as a man came up behind her.                                                      
No, not a man…..men.
There was one on both sides of her, the smell of beer tumbling off their tongues.
“Excuse me,” Feyre said, taking a step forward.
But a hand reached out, and grasped her wrist.
“I’ve seen you around campus,” the bigger one smiled. “You’re….You’re b-beautiful.”
“D-D-Don’t h-harass her,” the other smiled, predatorily. “She looks l-like a screamer.”
Her heart sped up, threatening to beat out of her chest. “Let me go.”
“Cute,” was all the latter one said, grazing a finger down her cheek. “Trust me, I’ll s-s-show you a-a good t-t-t-time.”
“Let her go.”
The pair stopped, and the bigger one slowly let go of Feyre’s wrist.
They turned around before she did, though she could sense their fear as she, too, turned to her savior.
The breath in Feyre’s throat caught at the sight of him.
He looked like – like the night, if the night could be a man. He didn’t seem much older than Feyre, in his all black clothing, perfectly fitted to his sculpted body. His hair was dark, too – the only thing Feyre was able to see under the dim reflection of the porch light were his deep, lavender eyes.
He was beautiful, in every sense of the word.
“You can step back, walk away, and never come within spitting distance of her again…..or I will report you, and you can spend the rest of your life begging for change.”
Feyre hadn’t even finished loosening her breath before they had sprinted off.
The stranger smiled, nodding his head toward the assholes. “Don’t worry. I’m reporting them anyway. Can I escort you somewhere, um….”
“Feyre,” she whispered, unsure of why her confidence had not returned. She couldn’t figure him out. Couldn’t pick up on why he had done what he had done when he looked like one of Lucifer’s own.
A dark, beautiful fallen angel.
“Hello, Feyre.” He stepped further into the light, his eyes becoming fierce with his devilish grin. “I’m Rhysand.”
Chapter 3 coming soon.
Let me know what you think!
If you wish to be tagged, ask in my ask box so I can keep track for the next chapter, please. :)
@wingedillyrian @throne-of-ashes-and-beauty @photofeesh @southern-by-gods-mercy @randomfanficshit
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mathematicianadda · 4 years
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Advanced Integral: $\int_0^1\frac{\text{Li}_2(x^2)\arcsin^2(x)}{x}dx$ https://ift.tt/eA8V8J
How to tackle
$$I=\int_0^1\frac{\text{Li}_2(x^2)\arcsin^2(x)}{x}dx\ ?$$
This integral came up while I was working on some harmonic series.
First attempt: By writing $\text{Li}_2(x^2)=-\int_0^1\frac{x^2\ln(y)}{1-x^2y}dy$ we have
$$I=-\int_0^1\ln(y)\left(\int_0^1\frac{x\arcsin^2(x)}{1-x^2y}dx\right)dy$$
and Mathematica gave a complicated expression for the inner integral and that made me stop.
Second attempt: $x=\sin\theta$
$$I=\int_0^{\pi/2}\theta^2\cot\theta\ \text{Li}_2(\sin^2\theta)d\theta$$
$$=\sum_{n=1}^\infty\frac{1}{n^2}\int_0^{\pi/2}\theta^2\cot\theta \sin^{2n}(\theta) d\theta$$
and I have no idea how to continue. Any suggestion?
Thanks
from Hot Weekly Questions - Mathematics Stack Exchange Ali Shather from Blogger https://ift.tt/327xlz0
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devphilamaths · 3 years
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Solve sin(pi/2+theta) | sin(pi/2+ x) | sin pi/2 + x formula, Find value sin pi by 2 + x
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devphilamaths · 3 years
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Solve sec(pi/2+theta) | sec(pi/2 +x) | sec pi/2 + x formula, Find Exact value sec pi by 2 + x
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devphilamaths · 3 years
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Solve cosec(pi/2+theta) | cosec pi/2 + x formula, csc (pi/2 +x) | Find value cosec pi by 2 + x
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mathematicianadda · 4 years
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Integrate a weighted Bessel function over the unit disk https://ift.tt/eA8V8J
I would like to evaluate a complex-valued integral of the form
$$ I_e = \int_0^1 x e^{iax} J_0(b \sqrt{1-x^2}) dx $$
where $a$ and $b$ are positive real numbers and $J_0(z)$ is the Bessel function of the first kind. I am particularly interested in the special case of $a=(c+1)b$ for $c \ll 1$.
The task boils down to evaluating two real-valued integrals
$$ I_s = \int_0^1 x \sin(ax) J_0(b \sqrt{1-x^2}) dx $$ $$ I_c = \int_0^1 x \cos(ax) J_0(b \sqrt{1-x^2}) dx $$
The integral with the sine has a simple form given by Gradshteyn and Ryzhik (6.738.1) which, after simplification, becomes
$$ I_s =\sqrt{\frac{a^2}{a^2 + b^2}}j_1(\sqrt{a^2 + b^2}) $$
where $j_1(z)$ is the spherical Bessel function of the first kind.
I am not exactly sure how this expression was derived. Perhaps it holds a clue. I tried substituting the integral form of the Bessel function and integrating analytically but did not get very far.
By symmetry, I naively expected the integral involving the cosine to be proportional to the spherical Bessel function of the second kind $y_1(z)$ (and thus, the complex-valued integral to be proportional to the spherical Hankel function of the second kind), but that does not appear to be the case.
$$ I_c \approx -\sqrt{\frac{a^2}{a^2 + b^2}}y_1(\sqrt{a^2 + b^2}) $$
The agreement for the special case of $a=b$ is not too terrible, but something is going wrong for small values of $b$. If I take the difference between the true value of the integral and my guess, it is proportional to another Bessel function, but the arguments are crazy and it does not make any sense.
Now, if I assume that it is proportional to the spherical Bessel function of the first kind instead, and compute the ratio between the true value of the integral and my guess, I get the classic graph of the cotangent.
For $b \gg 1$, it appears
$$ I_c \approx \cot(\sqrt{a^2 + b^2} + \phi)) \sqrt{\frac{a^2}{a^2 + b^2}}j_1(\sqrt{a^2 + b^2} + \phi) $$
In practice, this is hardly useful, since the period of the cotangent would have to correspond to zeros of the spherical Bessel function, and these are hard to compute. Additionally, $j_1(z)$ and the actual $I_c$ appear to be slightly off-phase.
Of course, ideally, I would like to solve the problem analytically, but I am not sure how (using a series expansion, perhaps?). I would appreciate any tips or guidance.
Thank you!
Edit: by transforming using $x=\sin{\theta}$, there is an almost perfect match in Gradshteyn and Ryzhik (6.688.2), except that the leading term in my case is $\sin{2\theta}$ rather than $\sin{\theta}$. Additionally, it gives a result in terms of the spherical Bessel function of the first kind rather than the second kind, so it does not explain my empirical approximation.
$$ I_c = \int_0^{\pi/2} \frac{1}{2} \sin{2\theta} \cos(a \cos{\theta}) J_0(b \sin{\theta}) d\theta $$
Using the representation given above, we can expand $\cos(a \cos{\theta})$ in a series, which does give an analytic solution, but it converges rather poorly and does not appear to contain any spherical Bessel terms (just the regular Bessel functions). I am yet to find an expansion in terms of spherical Bessel functions.
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mathematicianadda · 5 years
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$\int_{-\infty}^\infty \frac{e^{pz}}{e^z-1}dz$ Cauchy principal value
$$\int_{-\infty}^\infty \frac{e^{pz}}{e^z-1}dz$$
I started by defining the following contour: rectangular contour
It is easy to show that the integrals along the 2 vertical sides of the rectangle go to $0$ as $R\Rightarrow\infty$ by applying the triangle inequality for integrals. Note that we must have $0<p<1$ for this to work.
For the integral along the top side of the rectangle we define $z=x+\pi i$ so the integral becomes: $$\int_{R}^{-R} \frac{e^{p(x+\pi i)}}{e^{x+\pi i}-1}dx$$ Using some basic algebra and taking the limit as $R\Rightarrow\infty$ this simplifies to: $${e^{p \pi i}}\int_{-\infty}^{\infty} \frac{e^{px}}{e^{x}+1}dx$$ This integral has a known solution and plugging this solution in yields: $${e^{p \pi i}} {\frac{\pi}{\sin{p \pi}}}$$ Now we need to tackle the integral over the semi-circle and to do so we define: $z=\epsilon e^{i \theta}$ so $dz=i \epsilon e^{i \theta} d\theta$ and $-\pi \leq \theta \leq 0$
The integral becomes:
$$\int_{- \pi}^{0} \frac{e^{p \epsilon e^{i \theta}}}{e^{\epsilon e^{i \theta}}-1} i \epsilon e^{i \theta}d\theta$$ Taking the limit as $\epsilon \Rightarrow 0$ using l'Hopital's rule we get: $$\int_{- \pi}^{0} \frac{e^{p \epsilon e^{i \theta}}}{e^{\epsilon e^{i \theta}}-1} i \epsilon e^{i \theta}d\theta = i \pi$$ Putting everything together and using the fact that the closed contour integral is zero since there are no singularities inside the contour yields. I let WolframAlpha do the simplification:
$$\int_{-\infty}^\infty \frac{e^{pz}}{e^z-1}dz = -\pi \cot{(p \pi)} -2 \pi i$$ In my opinion this seems weird because the integrand is real for all inputs of z except $0$. I expected the cauchy principal value, which this integral actually is because it blows up at $0$, to be real valued as well. Unfortunately my calculations show that this integral is complex valued.
Could someone please tell me if I did something wrong or not. I found an article which also has this integral covered but I believe there is a mistake in their calculation of the integral along the semi-circle (they forgot a minus sign), which if corrected results in the same answer I got. link: article covering the integral.
Any help is greatly appreciated!
PS: I am 17 years old so this is my first post on this forum, if there are any (informal) rules which I am not aware of please let me know as well!
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mathematicianadda · 5 years
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Does $\sum_{k=1}^n|\cot \sqrt2\pi k|$ tends to $An\ln n$ as $n\to\infty$?
Question: How can we prove that $$L(n)=\sum_{k=1}^n\left|\cot \sqrt2\pi k\right|=\Theta(n\log n)$$ as $n\to\infty$? Furthermore, if $\sqrt2$ is replaced with a quadratic irrational number, does it still holds?
Numerical experiment. By plotting $$\frac1{n\ln n}\sum_{k=1}^n\left|\cot \sqrt2\pi k\right|,$$ we can find that it approximately tends to $0.6$.
Failed attempt of the upper bound. $$L(n)<\sum_{k=1}^nCk=C\frac{n(n+1)}2$$for some $C$. It can be easily deduced due to the irrationality measure $2$ of $\sqrt2$. Failed attempt of the lower bound. Asymptotically, half of the summand is greater than $1$ due to the irrationality of $\sqrt2$. Therefore, $L(n)>Dn$ for some $D$ when $n$ is large enough.
from Hot Weekly Questions - Mathematics Stack Exchange from Blogger http://bit.ly/2UIVzPh
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