## Definition

$\text{Л}(x) := -\int_0^x \log \left| 2 \sin t\right|\;dt$

Note that \begin{aligned} \text{Л}'(x) &= - \log |2 \sin x| \\ \text{Л}''(x) &= -\cot x \end{aligned}

The definition looks very strange. It's probably easiest to think of Л as being defined by the differential equation $\text{Л}''(x) = -\cot x$ subject to the initial conditions \begin{aligned} \text{Л}(0) &= 0\\ \text{Л}'\left(\frac \pi 2\right) &= 0\\ \end{aligned}

## Useful Facts

We begin with a seemingly-unrelated formula about the circumcircle of a triangle.

Let $t_{ijk}$ be a triangle with edge lengths $\ell_{ij}, \ell_{jk}, \ell_{ki}$ and angles $\alpha_{ij}^k, \alpha_{jk}^i, \alpha_{ki}^j$.
The radius of the circumcircle of $t_{ijk}$ is given by $R = \frac{\ell_{jk}}{2 \sin \alpha_{jk}^i}$.
Proof

We place our triangle inside of its circumcircle.

$c$ is the center of the circle. We draw a diagonal passing though $k$, and name the opposite point $r$. Note that angle $k-j-r$ must be a right angle, since the line $\overline{kr}$ is a diameter of the circle. We also place a point $s$ such that the angle $k-s-j$ is a right angle.

Note that angle $k-r-j$ must equal angle $\alpha_{jk}^i$ since both subtend the same arc between $k$ and $j$. Therefore, triangle $t_{isk}$ is similar to triangle $t_{rjk}$. In particular, this means that $\frac {\ell_{kr}}{\ell_{kj}} = \frac {\ell_{ik}}{\ell_{sk}}$ Note that $\ell_{sk} = \ell_{ik} \sin \alpha_{jk}^i$. Furthermore, $\ell_{rk} = 2R$. Thus, we see that $\frac {2R}{\ell_{kj}} = \frac {\ell_{ik}}{\ell_{ik} \sin \alpha_{jk}^i}$ Simplifying, we conclude that $R = \frac{\ell_{jk}}{2\sin\alpha_{jk}^i}$

### Energy Derivative

Futhermore, let $\lambda_{mn} = 2 \log \ell_{mn}$. We define a function $f( \lambda_{ij}, \lambda_{jk}, \lambda_{ki}) = \frac 12 \big( \alpha_{jk}^i \lambda_{jk} + \alpha_{ki}^j \lambda_{ki} + \alpha_{ij}^k \lambda_{ij}\big) + \text{Л}( \alpha_{jk}^i) + \text{Л}( \alpha_{ki}^j) + \text{Л}( \alpha_{ij}^k)$
$\pd f {\lambda_{jk}} = \frac 12 \alpha_{jk}^i$
Proof

Using the circumradius formula, this becomes a straightforward computation.

\begin{aligned} \pd f {\lambda_{jk}} &= \frac 12 \alpha_{jk}^i + \left(\frac 12\lambda_{jk} - \log |2 \sin \alpha_{jk}^i| \right) \pd {\alpha_{jk}^i}{\lambda_{jk}}\\ &\quad+ \left(\frac 12\lambda_{ki} - \log |2 \sin \alpha_{ki}^j| \right) \pd {\alpha_{ki}^j}{\lambda_{jk}} + \left(\frac 12 \lambda_{ij} - \log |2 \sin \alpha_{ij}^k| \right) \pd {\alpha_{ij}^k}{\lambda_{jk}}\\ &= \frac 12 \alpha_{jk}^i + \log\left(\frac {\ell_{jk}}{2 \sin \alpha_{jk}^i}\right) \pd {\alpha_{jk}^i}{\lambda_{jk}} + \log\left(\frac {\ell_{ki}}{2 \sin \alpha_{ki}^j}\right) \pd {\alpha_{ki}^j}{\lambda_{jk}} + \log\left(\frac {\ell_{ij}}{2 \sin \alpha_{ij}^k}\right) \pd {\alpha_{ij}^k}{\lambda_{jk}} \end{aligned} By the circumradius formula, each of those log terms is just $\log R$. (Equivalently, we could just use the law of sines here to observe that all of the terms are equal. It is not actually important what they are equal to). Thus, we have \begin{aligned} \pd f {\lambda_{jk}} &= \frac 12 \alpha_{jk}^i + \log R \pd{}{\lambda_{jk}} \left(\alpha_{jk}^i + \alpha_{ki}^j + \alpha_{ij}^k\right) \end{aligned} Since the sum of the angles in a triangle is always $\pi$, the derivative on the right vanishes. Thus, we obtain the desired equality \begin{aligned} \pd f {\lambda_{jk}} &= \frac 12 \alpha_{jk}^i \end{aligned}

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17. I was lost with no hope for my wife was cheating and had always got away with it because I did not know how or always too scared to pin anything on her. With the help a friend who recommended me to who help hack her phone, email, chat, sms and expose her for a cheater she is. I just want to say a big thank you to AlbaniaspyhackerAtGmailDotCom. I am sure someone out there is looking for how to solve his relationship problems, you
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20. I was able to recover my funds from a very sketchy company, 24Options, Last year a friend and i invested all our life savings but got duped in the process. Earlier this year, we were able to use the services of R E C O V E R C O I N @ R E S C U E T E A M . C O M and we have gotten all our money back. My nightmare is over, It's a whole new day here. Do be careful when dealing with investments,you can also reach them via call or whatsapp +1 775-372-9370.

21. It's a sad experience to lose your money to these wallets...I lost mine
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25. It got to my notice that when ever my girlfriend is close to me she reduce the brightness of her cell phone while texting, its been 2 months since i noticed this but i never wanted to ask her cause she would end up deleting all, so i did it my way by hiring an expert whose email is spyexpert0@gmail.com this hacker helped in breaking into her phone and i was able to read all that she had been hiding, she really thinks she was smart after my tour on her phone i saw a lot of things that i could not believe my girlfriend does because i trusted her so much shout out to  spyexpert0 the great.

26. I googled a few recovery companies and after a few days of research I came across a crypto expert who introduced me to a programmer that helped me recover my lost coins. I was very skeptical as can be imagined but I’m glad to say my bitcoin was recovered and sent back to me after an hour. You should file a complaint directly to him at (M O N E Y R E C O V E R Y 0 0 1 @ S O L U T I O N 4 U . C O M) in case u need help recovering your lost transactions, I recently recovered mine within an hour.

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manager telling me about my account and the bonuses I could get if I put in more money...he was
really good at convincing (maybe he brainwashed me ) I later put $400,000 in total. Then I sent all of my verification documents such as proof of address, ID and Bank card details (I sent without CVV) and Declaration form . Also I received lots of emails from them about verification and activation and etc. During the first 3 days which my account was funded and was ready to start; they called me many times to re-invest and increase my amount and I refused because they got too pushy about it. I figured something was wrong when I requested for a withdrawal and was told I had to reach a deposit target before I can make withdrawals. So I couldn't trust them as they changed their word and I asked the adviser I want to close my account as my account is verified and it's ready. I already told him that I can't put in more money and now I changed my mind and I want to withdraw my money and close my account. I send lots of email, chat, request, text, call and etc and ask them to call me and explain but nobody answers and replies to me even through email (Used to be emailed straight away) Now I want to withdraw all of my money but I can't go through my account. THEY NEVER CAME BACK TO ME. TEL, ADDRESS AND COMPANY AND ALL ARE SCAM. Until I was introduced to the guys at "CONTACT@ACTIVEBONORUM.COM" they helped me get my money back from these rippers after a long while...I owe my life to them right now. not everybody there would be fortunate to find a trusted recovery expert www.activebonorum.com 31. My experience with CryptoNx was completely unpleasant. They thought they would cease my hard earned profit after growing my account to over 120%. This started when I reached out to the account manager when I wasn't able to withdraw my money. He assured me a successful withdrawal within a week. 3 weeks went by and I was never able to reach him again. I knew I'd been scammed. So I started researching a credible wealth recovery agency that could get my funds as soon as possible without any stressful uncertain chargeback procedures. The best decision I ever made was to work with Sharon and his team from " activebonorum.com ". Unbelievable, They got penetration testers that recovered my funds including my profits in less than 10 days. Guys! I couldn't be happier. Please stay clear of these unregulated brokers. Email: contact@activebonorum.com 32. If you're going through issues with your credit report and upgrading your credit score, I will advise you to contact CLEVERHACKER.HACK@GMAIL.COM . He's capable of by passing into the 3 credit bureaus. He helped me delete an old eviction on my credit report and help me increase my credit score to 794. He has done a great job for me and i can share that he's a professional at what he does. You can also contact him directly on his number (803) 8145462. Thank me later 33. Please beware there are many fake cryptocurrrency investment platforms out there, I got scammed twice before I met a genius hacker who helped retrieve my bitcoin. The whole plan was so smooth I could not doubt it. Bitcoin is actually a great investment option but one thing I discovered over time is that it is not possible to mine bitcoin so don’t be deceived. I invested 15BTC on a particular website called FoxBtc, I monitored the profit yielding but got locked out of the account before I could withdraw. I went online and met this recovery expert at recoverywealthnow360 on google mail who helped me retrieve my bitcoins back in few hours. He saved my life , he can save yours. 34. I was able to get the best financial aid of these knowledgeable computer analyst, I got my university degree and right now i'm working in one of the best company in Pennsylvania, kindly contact EAGLESPYHACKER@GMAIL.COM for any of your services and I assure you that they will work to your satisfaction, I can boldly say they are the best cyber gurus, because I have witnessed their good work, which is why I am recommending them to you now Their services includes * They make bank transfers within an hour without traces * BITCOIN transfer within minutes * Sales of University degree titles * They recover Money back from scammers * Tracking of victim's locations without complications * Loading credit card account for you within an hour without traces * Whats-App spying within 45 minutes * Bank transfer within 2 hours * Creation of DRIVER'S LICENSE * Fixing of credit score records and lots more * Loan Offers * Sales of Blank ATM and Credit Cards * clearing of bad driving records and criminal records * Help to register at ILUMINATI and become richer * Change of university degrees * Spying of all social media account within * Sales of university Titles originals. * We make the accreditation of account without a trace *Clearing of bank debts within 2hours contact them now for any of your service and thank me later EAGLESPYHACKER@GMAIL.COM 35. Thank you jeajamhacker@gmail.com 36. Please beware there are many fake bitcoin mining/clouding out there, I have been a victim once but now a certified blockchain consultant and I know better. The whole plan was so smooth I could not doubt it. Bitcoin is actually a great investment option but one thing I discovered over time is that it is not possible to mine bitcoin so don’t be deceived. I invested$25000 on a particular website called eurekaminingblock, I monitored the profit yielding but was told to open a new blockchain account to receive my payout. A public wallet was imported into the account and I was made to believe that was my profit. The bitcoin was labelled non spendable and it took me 2 years to be able to access it without the knowledge of the company. The non spendable bitcoin is the scam out there now and a lot of people are falling victim of it. I found Mr Dawson Wright on Quora who helped me access my investment together with the profit without the knowledge of the company. You can reach out to him also on fundsretrieval1 @ gmail . com if you need help, he is genius. Goodluck!
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38. FRESH AND UPDATED STUFFS WITH 98% VALIDITY RATE ( DUMPS/ DUMPS+ATM PINS /CVV & fULLZ /, CC TOP UPS, BANK LOGS, VERIZON, PAYPAL ACCOUNTS, EGIFT CARDS FOR WALMART AND AMAZON AND SPAMMING TOOLS FOR SALE) via Email: atmgeniuslinks@gmail.com / ICQ: https://icq.im/Anouymous_CC / Telegram: @Anonymous_Ccs

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..................................................................

ALL COUNTRY CREDIT CARDS and fullz...........

CCnum:: 4400663873545159
Cvv: 094
Expm: 06
Expy: 19 Fname: Gary
Lname: Hoskins
City: MILFORD
State: OH
Zip: 45150
Country: USA
Phone: 5138318649
Email: gch619@fuse.net

DEBIT : 432190 – 99, 445808, 450644, 450670, 451948, 454003, 458501, 471401, 472476, 486342, 503886, 522122, 528030, 529558, 531704, 531729, 539847, 545409, 551029

FULLZ
'''''''''''''
First Name : Kathryn
Middle Name : M
Last Name : Ratledge
Spouse Name : Ruth Ratledge
Father Name : Thomas Ratledge
Billing Address : 9 Ridgewood Dr
City : Greenville
State : CA
Zip Code : 29615
Country : US
Phone Number : 15629234492
Credit Card Information :
*********
Card Type : Credit
Credit Card Number : 4737033009002349
Exp. Date : 02/23
Name On Card : Kathryn Ratledge
Cvv2 : 736
ATM Pin: 5910
Bank Name : Bank of America
Routing Number : 122000661
Account Number: 0222301305
Mother Maiden Name : Coleman
Social Security Number : 340224188
Birthday : 19
Birth Month : 8/June
Birth Year : 1980
Account Information :
*******
AOL ID : Glendeweese

Account Open In: LD
Online ID: GROVANS11x
ATM or Check Card PIN: 001455
Passcode: eg17

......................................................................
DUMPS WITH PIN
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Track1 : B4784559000586171^Andrew/Mayerbach^190310100000000000003343300003700000000
Track2 : 4784559000586171=19031010003343300003700
ATM Pin : 5241
------

Dumps + Pin ( Canada )
Track1 : B4530920124710013^Bibliotheque^2203220612800000054 7000000
Track 2 : 4530920124710013=22032206128000547000
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39. Today Thursday 5th 2020 I found out about my cheating husband with the help of jeajamhacker@gmail.com

40. It can be stressful seeing as bitcoin is crypto currency but it is possible. In recent times, the tables have turned for internet scammers who think they can do what they like on the internet and get away with it. This is why I want to implore you to start a chargeback and reach out to R E C O V E R C O I N @ R E S C U E T E A M . C O M. they have an elite forensic team and agents on standby available to help you in your recovery process.

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42. I noticed that my husband is having multiple affairs but I was never sure because I needed proof of what has been going on and am also happy I got the proof I have always wanted  with the help of jeajamhacker@gmail.com I got to access my husband phone remotely and that was how I knew that my thought has been right about my husband but didn't have any one to hack his phone. A very big thank you to the world for referring me to use jeajamhacker@gmail.com services and also with their good reviews about jeajamhacker@gmail.com I was moved to email jeajamhacker@gmail.con for help!

43. Have you ever gotten your bitcoins stolen from your wallet or invested in an ICO that turned out to be a scam, you are not alone because this happened to me too. I initially lost $90,000 in just 3 months from Cryptoallday, I contacted the authorities and they referred me to a bitcoin recovery company who helped me recover all my funds within less than 2 days... I'm speaking out to improve our awareness of these cryptocurrency thieves and help you if you have been a victim yourself. Simply send them an email bitcoinretrieval2018 gmail com 44. Be careful and stay away from binary trading if you do not know of a way to get your investments back after losing it, tried different recovery service providers and at the end I lost more than I invested, but at BITCOIN RECOVERY CONSULTANT, they can help you trade and still help you recover your investments from the broker you lost to, this is how I get my investments back so I have no worries because B T C R E C O V E R Y 1 0 1 @ C O N S U L T A N T . C O M, they have your back and make sure you never lose again. 45. russiancyberhackers@gmail.com always the best hacker to email when it comes to hacking a cheating spouse cell phone. 46. hello guys,i have just used hacking service of emrysghosthacker on gmal..i think this is the solution everyone out there facing any hacking related issue deserves, after been ripped of by many hackers out there, i now finally got a solution to my long time cheating husband’s phone..i can now view all his call logs, text, fb and all other social media conversations directly from my own phone..i recommend this guy as the best option now, his fast and reliable cuz he kept his promise with the information delivery and i promised him to share his reputation across online for getting this done for me atlast.Show less 47. Hello Everyone, Let me share this with you, My Husband was good and smooth at hiding his infidelity and cheating lifestyle so i had no proof for Years, I was referred to this great ethical hacker and private investigator and i decided to give it try. Guys the result i got from this Man was incredible because all my cheating husbands day to day activities, the deleted messages, Text messages, whatsapp, facebook, instagram, Email and even phone conversations was wired directly to my cell phone. Thanks Man for being honesty and truthful in your job delivery.. I promised to always talk about your good professional and quality job delivery online for anyone who has the same problem and needed the same help and also introduce more people to you always...To everyone reading this If your spouse is an expert at hiding his cheating adventures Contact this Honest , Fast and trusted hacker to do justice for you Via.... 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No matter what life brings, you have to be bold enough to confront your cheating spouse with evidence I give thanks a friend who intoduced me to the great hack of all time who helped get all secret files my wife have been hiding for several years of marriage he helped hack and access into my cheating wife social networks, icloud and much more, viber chats, Facebook messages and yahoo messenger, calls log and spy call recording, monitoring SMS text messages remotely, cell phone GPS location tracking, spy on WhatsApp Messages, gmail and kik and I got to know that she was cheating on me , in less than 8 hours within this sort period I was able to access her phone through mine he helped me out, contact him if you have any issue similar { spygodr @ g m a i l. c o m OR you can call or text him on +1 518 722 4413 his work is discreet and delivers. Tell him I referred you. 49. Greetings.... Check out these credit cards today. My name is Max Bradford from California. 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Bitcoin Private key hacks and tips to learn large amounts of Bitcoins faster Start codding and hacking today for free follow out assistance and do your own hack with just a couple of directions. spend non spendable funds, send fake transactions How to get free bitcoins, Bitcoin non spendable hack,private key hack,bitcoin generator. recover your lost funds today, We will give you a step to step guide how to recover this funds in blockchain. website: http://spygodhack.com/ Email: recbtc17@gmail.com Phone/ Whatsapp: +1(518)722-4413 51. I was scammed by a binary option website iq options to be exact. I lost about$153,000 to them. It was a really hard time for me because that was all i had and they tricked me into investing the money with a guarantee that i will make profit from the investment.. I was referred to Mr Gary, he’s a senior recovery specialist and he helped in recovering all my lost funds to me. I couldn’t believe at first that this was possible as i have lost all hope in contacting anyone on the internet, but i guess not everyone is bad after all. You can reach him on cybertechhub100 at gmail dot com

52. I was scammed by a binary option website iq options to be exact. I lost about $250,000 to them. It was a really hard time for me because that was all i had and they tricked me into investing the money with a guarantee that i will make profit from the investment.. I was referred to Mr Gary, he’s a senior recovery specialist and he helped in recovering all my lost funds to me. I couldn’t believe at first that this was possible as i have lost all hope in contacting anyone on the internet, but i guess not everyone is bad after all. You can reach him on cyberhelp.me22at gmail dot com 53. Are you interested in the service of a hacker to get into a phone, facebook account, snapchat, Instagram, yahoo, Whatsapp, get verified on any social network account, increase your followers by any amount, bank wire and bank transfer. I can vouch for him because I have used him to monitor my husband many time when I feel suspicious about his movements TALENTED HACKERS DO YOU REQUIRE A CERTIFIED HACKER FOR : +University grades hack, +Bank account hacks, +Control devices remotely hack, +Facebook Hacking Tricks, +Gmail, AOL, Yahoomail, inbox, hack are available, +Database hacking, +PC computer tricks +Bank transfer, Western Union, money gram, Credit Card transfer +Wiping of credit, +VPN software, +ATM Hack we are the real deal when it comes to hacking, carding, WU transfer, Money Gram transfer etc contact us now hackintechnology@gmail.com ‪+1 (213) 295‑1376‬(whatsapp) 54. Obrigado por compartilhar! Não conhecia todos, mas gostei de algumas mudanças, como opção por diferentes visualizações como espionar facebook 55. Welcome. BE NOT TROUBLED anymore. you’re at the right place. Nothing like having trustworthy hackers. have you lost money before or bitcoins and are looking for a hacker to get your money back? You should contact us right away. It's very affordable and we give guarantees to our clients. Our hacking services are as follows:Email:Creditcards.atm@gmail.com -hack into any kind of phone _Increase Credit Scores _western union, bitcoin and money gram hacking _criminal records deletion_BLANK ATM/CREDIT CARDS _Hacking of phones(that of your spouse, boss, friends, and see whatever is being discussed behind your back) _Security system hacking...and so much more. Contact THEM now and get whatever you want at Email:Creditcards.atm@gmail.com whats app:+1(305) 330-3282 WHY WOULD YOU NEED TO HIRE A HACKER??: There are so many Reasons why people need to hire a hacker, It might be to Hack a Websites to deface information, retrieve information, edit information or give you admin access. • Some people might need us To Hack Their Target Smartphone so that they could get access to all activities on the phone like , text messages , call logs , Social media Apps and other information • Some might need to Hack a Facebook , gmail, Instagram , twitter and other social media Accounts, • Also Some Individuals might want to Track someone else's Location probably for investigation cases • Some might need Us to Hack into the Court's Database to Clear criminal records. • However, Some People Might Have Lost So Much Funds With BINARY OPTIONS BROKERS or BTC MINING and wish to Recover Their Funds • All these Are what we can get Done Asap With The Help Of Our Root Hack Tools, Special Hack Tools and Our Technical Hacking Strategies Which Surpasses All Other Hackers. ★ OUR SPECIAL SERVICES WE OFFER ARE: * RECOVERY OF LOST FUNDS ON BINARY OPTIONS * Credit Cards Loading {Any country} * BANK Account Loading {Any country} ★ You can also contact us for other Cyber Attacks And Hijackings, we do All ★ ★ CONTACTS: * For Binary Options Recovery,feel free to contact (Creditcards.atm@gmail.com)for a wonderful job well done,stay safe. Why waste your time waiting for a monthly salary. When you can make up to$3,000 in 5-7days from home,
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57. HOW TO RECOVER LOST BTC I lost my bitcoin to fake blockchain.com impostors on Facebook. Hello I have to share my experience with a life saving group called alliedrecoverygroup from some blockchain scammers.
They contacted me as blockchain official support and I fell stupidly for their mischievous act, this made them gain access into my blockchain wallet account, whereby 7.0938 btc was stolen from my wallet in total.
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spending limit in stores and unlimited on POS.

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**BEWARE OF SCAMMERS AND FAKE HACKERS IMPERSONATING US BUT THEY ARE NOT
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I major in…
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Apps hacking
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Regards....

62. Are you a victim of ;

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