Tuesday, February 23, 2021

Abstract Algebra Problem Solution

  Pr0blem:  Prepare the respective tables for the isomorphism and give specific examples in terms of the function φ, i.e. show specific mappings. (Where: φ(x) φ(y) = φ(xy) for example).

Solution:

The solution included preparation of the respective addition table (in  Z4 ) and multiplication table (in  Z10 ). The addition table for Z4 is (see e.g. Fig. 1 in http://www.brane-space.blogspot.com/2012/11/looking-again-at-groups-sub-groups.html ):



+ /----0 ----1 ---2 ----3
-----------------------
0 --- 0------1 ---2 ----3

1 -----1 -----2 ---3---- 0

2 -----2 -----3--- 0 ----1

3----- 3 -----0 ---1---- 2
-----------------------------


The multiplication table in Z10


-*--/----6 ----2 -----4 ----8
---------------------------
6 --- ---6------2 --- 4 ----8

2 ------2 ------4 ----8----6

4 ------4 ------8--- -6 ---2

8----- 8 -----6 ----2---- -4
-----------------------------


Since 6 *6 = 6 then the identity element in  Z10  is 6.. 

So φ(x) φ(x) = φ(x^2) = (φ6)φ(6) =φ( I)

Then the mapping isomorphisms between the tables will be:


Φ: 0 -> 6 (identity to identity element)

Φ: 2 -> 4

Φ: 3 -> 8

Φ: 1 -> 2


So each of the numbers on the left side of the arrow is mapped to the corresponding number on the right side, and this is done using the two tables such that each element of Z4  is mapped to each element of Z10.




-----------------------------

Seriously? Rush Limbaugh's Odious Talk Show To Be Revived Using "Decades" of Archived Rot?



                    Limbaugh (top) at his FLA estate and my late brother Mike- a dedicated Ditto head and gun collector.

The notion that Rush Limbaugh's brand of vitriol, hate and division didn't end when he did, is appalling in and of itself.  But now we learn ('Limbaugh Show Will Continue Using Archives',   WSJ,  Business Exchange, Feb. 18-20, p, B5) that this bombastic buffoon's execrable, hate-filled drivel will survive him, given that according to Premiere Networks (his distributor)  the Rush Limbaugh Show: "Will continue using archived segments and clips until his audience is prepared to say goodbye."  

 Yeah, but hogs that love rolling in shit never tire of that pastime, so it's doubtful the Dittoheads will be any different and "tire" of recycled Limbaugh talk radio hog shit.

Wasn't it enough that the name of Rush Limbaugh, nee "El Rushbo", a.k.a. the human swine, will live forever in infamy after decades?   Does Premiere Networks really need to resurrect Limbaugh's hundred thousand hours worth of daily insults, racist putdowns, attacks on women, the disabled and even children traumatized by school shootings?  Who can forget-  some 8 years ago -  the odious spectacle of this slobbering hog imitating kids crying in fear of another school shooting?   El Rushbo sputtered in a mocking, sneering voice, "Awww, they don't wanna  die...", this after the Sandy Hook slaughter that saw 20 school kids gunned down in their classrooms.   To Rush it was just fodder for his ridicule and the amusement of his deplorable dittoheads.   

Then there was the savage attack on Sandra Fluke who - in a March, 2012 appearance before a House Committee - e.g.

made a strong case that Georgetown University should provide contraception to students.  She related the enormous costs imposed on students as well as employees using contraceptives for health-prophylactic purposes. And  how withholding this service made it prohibitively expensive - even relating how it led to a friend having to undergo a surgical ovary removal in order to treat an ovarian cyst.  This when ordinary contraceptive use would have obviated the need for surgical intervention, risks of general anesthesia etc.

 And what did the cigar- smoking swine do? He launched a 3-day attack on her character, including barfing slanderous bollocks, e.g.

"It makes her a slut, right? She wants to be paid to have sex. She's having so much sex she can't afford the contraception. She wants you and me and the taxpayers to pay her to have sex,"

After which his diatribe continued as his gurgling lowbrow apes reveled in the procession of bon mots. Especially after he joked that he'd:  "buy these women all the aspirin they want to hold between their legs."   Making his audience of misogynistic Mandrills howl with glee.   But as I pointed out in a March 3, 2012 post: 

 "This fat,  slobbering pig clearly didn't grasp the biology of women in conjunction with the use of contraceptives.  Which had to be taken every day to be effective, unlike the Viagra El Rushbo was on (and busted for carrying undeclared  into a Dominican airport some years ago)."    

To its credit, Georgetown defended Ms. Fluke, calling the comments of Limbaugh "misogynistic, vitriolic and a misrepresentation of the position of our student."  

Following Georgetown's intervention on Sandra's behalf the predatory porker issued a halfhearted apology as advertisers fled his show.   In fact, they ought to have fled his 3-hour daily parade of verbal vomit years earlier.  This after the swine mocked actor Michael J. Fox - who suffers from Parkinson's disease-  for his appearance in a campaign ad. In one instalment, the bloated toad mocked Fox's tremors, croaking:  

"He is moving all around and shaking and this is purely an act. I have never seen Michael J. Fox display any of these symptoms of the disease. He can control himself enough to stay in the frame of the picture and he can control himself enough to keep his eyes on the lens, the teleprompter. But his head and shoulders are moving all over the place. This is really shameless, folks. Either he didn’t take his medication or he’s acting.”

The crudeness speaks for itself and is redolent of Trump and his own attack on a disabled reporter during his 2016 campaign. As one 'Daily Beast' contributor put it:  

"Is it no surprise that, a decade later, a political candidate would mock a journalist’s disability at a rally and the act would help him win the presidency?"

As the DB  writer went on to aver:  "Accusing the actor of faking his Parkinson’s symptoms, all because he chose to appear in an ad supporting someone on the other side of the political aisle from Limbaugh, isn’t just obviously heinous, its patronizing."

But this is the Limbaugh many had come to know and why he had built up such a reservoir of hate on the left.  Including for his incessant downplaying of Covid 19 as "just another kind of flu" -- thereby directly contributing to a significant fraction of the 500  thousand plus Covid deaths thus far. Contrary to Dennis Prager,  another Right wing screech monkey ('The Left Hated Rush Limbaugh Because He Had Their Number', Feb. 18, p. A15) that he "wasn't a mean -spirited man".    Only "sometimes" getting carried away- as with Ms. Fluke- and then, well he did apologize.  The truth?  The left had this bellicose Bozo's number from the get go and never let up.

 Including his penchant for crass opportunism, who'd stoop to naysaying the virus to grab more listeners - never mind many ending up 6 ft. under from following his advice.  Which makes one wonder what alternate universe Prager inhabits, given that Limburger paved the way for the incivility and polarization Trump used to gain ascendancy.  And in the process drag the nation through a sewer of degraded politics and debased civic space from which it may take years to recover  - if we ever do. 

 At least WSJ columnist Peggy Noonan ('Rush Limbaugh's Complicated Legacy', Feb. 20-21, p. A13) -  though at the outset she lets a dotty sentimentality blind her to many of his character defects -  finally admits that appeasement of his "dittoheads"  led to political polarization and a measure of dishonesty.  She is also smart enough to see that Reagan's repeal of the Fairness Doctrine is basically what turned a humdrum talk show host into a literal Reich wing Kraken  - poisoning the U.S. political landscape.  And setting the stage for the subsequent propaganda blitz and fake news torrent from FOX News.

Recall that Fairness Doctrine, starting in 1948, mandated that holders of public broadcast licenses give airtime to both sides of important issues.   Thus, given public property was being used it established a real public interest obligation from broadcasters.  Its repeal let loose the furies of hell and the ensuing downward slide in civility as Noonan admits, e.g.  "In 1987 the doctrine was abolished, but I don't know...  Has anything in our political culture gotten better since it was removed?  Aren't things more polarized, more bitter, less stable?  I'm not sure it was good for America."  

Well, indeed, it wasn't "good" - meaning advancing the nation's interests- because it allowed distortions, lies and propaganda to be propagated with impunity in the name of perverted free speech.  It basically set the stage for the mendacity of Trump and culminating in his seditious efforts to overthrow a duly elected government with an insurrection. 

 In a  real sense, Limbaugh and his multitude of Reich wing clones - like Dennis Prager- are responsible not only for the ongoing degradation in civility but of cratering Americans' critical thinking faculties.  Including lowering mean I.Q.  The fact millions of dittoheads remain glued to the repetitive grievance machine alone, testifies to the unlikelihood of ever reading a book, or even a newspaper. Hence, of  increasing their mental acuity by even one erg of energy.

See Also:

by Amanda Marcotte | February 19, 2021 - 8:04am | permalink

AND:

by Heather Digby Parton | February 21, 2021 - 7:36am | permalink

by Jaime O'Neill | February 20, 2021 - 7:37am | permalink

Excerpt:

As sad as it surely is to those legions of Rush Limbaugh fans across the nation, there is a silver lining to be found in this very dark cloud. It is a maxim long and oft repeated by people who believe in American initiative and enterprise that the way to success is to "find a need and fill it." Well, Rush Limbaugh was paid $85 million a year to satisfy a need lots of Americans had for a steady radio diet of hatred, bigotry, and bad taste.


Monday, February 22, 2021

Buzz Kill: Day Traders May Be In For Some Nasty Shocks From The Tax Man

 


As we read the news (WSJ, Feb. 16, p. B14) that the stock of the video game retailer Game Stop has fallen 80 percent from its late January peak, one wonders what tax hits await the armchair traders.  Will they be in the form of dozens of pages of 1099 tax documents - such as hit a 38 yr. old day trader named Joseph Holler (WSJ, 'Day Traders, Here's Your Tax Bill', Feb. 6-7, p. B5)?  Or will the shock be in the form of actual taxable hits?

Oh yeah, these armchair day traders - including a lot of kids (needing a new activity apart  from video games) -  have had a lot of fun.  Beats playing  Fortnite for 12 straight months -  or computer chess -  but now reality is about to pay a visit.  One thing for certain, newly minted traders will wake up to the tax effects of their hobby in the coming weeks as they receive bundles of tax forms.  In the words of the guy Holler, cited earlier:  "It's so convoluted I have no idea what it means. I think I will need to hire a professional!"

Ah well.  What are some of the other aspects the day trading hobbyists might need to learn?

The good news?  No Social Security or Medicare FICA taxes are due on this income.  But, there's also no tax withholding when selling a winner, say like when an employer sends part of your paycheck to the IRS.  What happens is that your broker,  commission paid or no,  sends each sale to the IRS even if the gain is only a few cents.  (This is how Mr. Holler will end up with over 200 pages of 1099 tax documents). 

 The takeaway here? Day traders who sell winners need to set aside cash or have a strategy for paying (estimated) taxes when they come due.   Quarterly payments may well be required and there's a penalty for noncompliance.   

More good news:  Those traders who hold investments longer than one year before selling are eligible for favorable tax rates of up to 23.8 % on their profits, known as capital gains. (Most day traders don't qualify because they sell within a year - so both gains and losses are classed as "short term".  

 The bad news for the short term  sellers is their net gains are taxed at higher rates, e.g. for ordinary income such as wages. These higher rates can range from 10 % to 37% - plus 3.8 % surtax for higher earners.   Then there is the state tax which may be due as well.  If the day trader happens to live in California he or she may be looking at a top income tax rate of 13.3% with no reduced rate for capital gains.  A top bracket frequent trader could conceivably see one half his gains going to federal and state taxes.  

As for losses. there have been a lot of them - say for GameStop traders- in the past month.   The good news is that the tax code allows traders to put losses to good use by offsetting capital gains. So if a traders has $10,000 of capital gains and $9,000 in losses after sales - in the same calendar year- then this yields a net taxable gain of $1,000.  Conversely, $10,000 in losses and $9,0000 in gains amounts to a net loss of $1,000.    Here it's also useful to know that unused losses can offset up to $3,000 of income such as wages.  Any balance carries forward to reduce future gains, or $3,000 in ordinary income a year.  

 One little detail that often snags armchair day traders concerns timing.  Thus, capital losses - say in your GameStop shares-  won't offset gains if the trader buys the same holding within 30 days before or after selling at a loss.  That's even if the stock is in a different account.  (This is called a "wash sale".)   Wash sale rules don't apply to stocks sold at a gain.  So if a trader sells a loser with a $10 loss per share and a winner with a $10 gain per share - in the same year- the trader can rebuy the winner right away.  

 Timing matters from year to year, again taking GameStop shares as an example.  Say a GameStop trader had sales last year- say at the end - and racked up $12,000 in net gains.  Then as GS shares dived early this year the trader sold the losses - and those amounted to $12, 500.Alas, these losses can't carry back to offset the 2020 gains - so the trader is stuck with that tax bill on the $12,000 net gains.   What about the 2021 losses?  Well they can shelter future gains, just not the previous year's.  

 Lastly, there is frequent trader status to consider.  The benefits for claiming this status include:

- deducting the cost of a home office, including special terminals

- deducting business meals, tax prep and other expenses  all on Schedule C

To be eligible - according to general iRS current rules (which have yet to be specifically detailed by the IRS), the trader wold need to rack up at least 4 hours a day of trading for 4 days a week in a given year.   Oh, and make more than 720 trades.

Like to trade a lot, day to day, to break the lockdown monotony?  One buzz kill is that soon the army of Game Stop (and other) day traders will receive a Form 1099 -B listing all these trades they may have forgotten. But when they receive hundreds of pages for the 1099-B they will know they haven't.  

Got $1,000 in cash left in your Game Stop account?  You may need it for the tax preparer.   According to one Denver tax accountant:  "If you've been day trading don't assume you can do your taxes yourself."

Yep, day trading can be a spiffy way to break the monotony of an ongoing pandemic, but be assured the tax man can make things even more "interesting".  





























 terms of tax obligations.   Well, what are some that the young trading Turks need to be mindful of?

Friday, February 19, 2021

Considering A Classic Abstract Algebra Problem

 Let (G, o) and (H, o) be groups. Then a homomorphism of (G, o) into (H, o) is a map of the sets G and H which has the following property:  f(x o y) = f(x) o f(y)


Example:

(G, o) = (R1 +)

(H, o) = (R*, ·)

Take f = the exponential function, so f(x) = exp (x), f(y) = exp(y)

Then: f(x + y) = exp(x + y) = exp(x) exp(y) = f(x) f(y)

Or:

H = R* = {x Î R: x  not equal 0}

And: exp R -> R* so exp(x + y) = exp(x) exp(y)


Def.: Isomorphism: An isomorphism of G onto H [(G, o), (H, o)] is a bijective homomorphism.

Example: H = P = {x Î R: x > 0}         (P, x)

Let G = (0, 1, 2, 3) for the operation (o) which is addition in Z4

Let H = (2, 4, 6, 8) for the operation (o) which is multiplication in Z10

Problem: Prepare the respective tables for the relevant isomorphism and give specific examples in terms of the function φ, i.e. show specific mappings.  (Where: φ(x) φ(y) = φ(xy) for example)


Hint: Check out this earlier blog post on isomorphisms: 



Perseverance Rover Spacecraft Touches Down On Mars After "Seven Minutes of Terror"

           Mars surface visible from Perseverance engineering camera.


 NASA's science vehicle Perseverance, the most advanced astrobiology laboratory ever sent to another world, streaked through the Martian atmosphere on Thursday and landed safely on the floor of a vast crater,  as Janice and I watched events unfold in real time.  We joined the NASA channel on Youtube in real time, just as the notorious "seven minutes of terror" had begun.  That designation owing to the final minutes during which the craft had to decelerate from hundreds of kilometers per second, to a few then zero.   But JPL engineer and flight controller Swati Mohan had the dynamics well in hand when she announced final touchdown on the Martian surface.  This quickly followed up by the image of the landing area shown above.

Mission managers at the jet propulsion laboratory burst into applause and cheers as radio signals confirmed that the six-wheeled rover had survived its perilous descent and arrived within its target zone inside Jezero crater, the site of a long-vanished Martian lake bed.

The robotic vehicle had sailed through space for nearly seven months, covering 293 million miles before piercing the Martian atmosphere at 12,000mph (19,000km/h) to begin its approach to touchdown on the planet’s surface.

The self-guided descent and landing during a complex series of maneuvers had been dubbed “the seven minutes of terror” and stands as the most elaborate and challenging feat in the annals of robotic spaceflight.  With successful touchdown at the end of the 7 minutes, engineer Swati Mohan announced:

“Touchdown confirmed! Perseverance safely on the surface of Mars, ready to begin seeking signs of past life,”

As a second round of cheers and applause erupted in the control room -  as the images of the surface arrived -  Janice looked at me and remarked: "That is one smart woman!"   I told her of that there was no doubt, adding: "She's definitely Mensa material and maybe Intertel too."

As evident from the top image, the scene was partially obscured by a dust cover.  It is a view from one of the Perseverance’s hazard cameras, showing the flat, rocky surface of the Jezero crater.

Perseverance is full of firsts. The first ever search for signs of ancient life on Mars. The first helicopter ("Ingenuity") to fly on another planet. The first recordings of sound on the red planet.  These "firsts" aside, the craft - about the size of a small car- will have a packed agenda the next few years.  The plans include exploring the Jezero crater - the site of an ancient lake that existed 3.9 billion years ago.  It will also search for micro-fossils in the rocks and soil.   

As for the Ingenuity helicopter excursions, these will be limited at first, especially as the aerodynamics still need to be tested in the thin Martian air (only about 1 percent the density of Earth's).  To compensate for the thin air, the mini chopper will spin its rotor blades at thousands of times per minute in order to generate a Bernoulli (lift) effect,  Plans are to have the bird fly no more than 70 m or so for the first few junkets.   

Meanwhile, most space junkies are looking forward to having the first pristine samples of Martian soil and rocks returned to Earth- sometime in the 2030s.  According to Thomas Zurbuchen, associate administrator for NASA's Science Mission Directorate:

"Perseverance is the first step in bringing back rock and regolith from Mars!"
 That sounds really cool, but make sure you sterilize that regolith first to remove any Martian microbes.   We do not want a pandemic of Martian origin.
See Also:
https://www.jpl.nasa.gov/

Thursday, February 18, 2021

'Freezing Fiasco' In Texas: But Don't Be Dumb Enough To Blame "Green Energy" Lapses

 "This is not the breakdown of the system, this is a system that is broken down by design.  It's designed not to incentivize weatherization of these facilities, and not to be prepared for a cold weather event such as we had this past week. On top of that a lot of people will get the highest electricity bill they've ever had when this is done."  Former San Antonio TX mayor, Julian Castro on ALL In last night.

As headlines scream about the catastrophe  unfolding in Texas with over 2 million still without power, some salient facts may have been missed.  For example, that the state has one of the most deregulated energy markets in the nation.  This in turn has spawned a system simply not designed to survive days of freezing temperatures. 

At the center of the events we find the single  Texas grid operator (ERCOT):   the Electric Reliability Council of Texas .  It encompasses one control area, hundreds of infrastructure owners and lots of coordination to make it work.

Aside:  There are, in the contiguous United States, three major interconnected systems — one covering everything east of the Rocky Mountains (the Eastern Interconnection), one for everything west of the Rocky Mountains (the   Western Interconnection), and  one for Texas- governed by ERCOT.   The power system that serves 95 percent of the state is intentionally isolated from the rest of the country.   This has been deliberate given a severely deregulated energy system is incongruent with regulated ones.  In this case Texas features a competitive wholesale power market but which offers scant incentives for investment in backup power. 

Ultimately, this outage, like many of the biggest blackouts before it, reflects the challenge of unanticipated events and consequences.  But it also reflects a "wholesale" market system that- in order to keep costs generally cheap- plays the margins and has deregulated to an astounding extent.  Among the benefits, the state could set goals, foster investment and expand transmission without input from other state or federal agencies.   But the downsides of the cheap energy model include NO incentive to winterize the infrastructure - whether insulating windmill turbines, or natural gas pipelines, etc.

 Thus, the  gamble taken was only advantageous until a "black swan" event occurred - such as the expanding polar vortex this past week.   The Texas system then had "bet the house" that saving investment money - by not winterizing generators, not insulating them, might work - and lost that bet.  On paper, in theory at least, it might have looked solid given Texas generally does not have winter storms like this one.  But when it did arrive, all bets were lost on any residual  resilience in the ERCOT system. As the frigid temperatures set in Texas ERCOT discovered natural gas pressure was not adequate to get this energy to its destinations, and outages ensued   People found their power gone, heat gone, and even water off since it depends on electricity.   Many became so desperate they had to chop up backyard fences for wood to burn in fireplaces.

At the same time one beheld Texas Governor Greg Abbott appearing on FOX news and blathering about a failure in the ERCOT system's wind and solar components as being the culprits.  As he put it:  "As you saw our wind and solar got shut down and they were collectively more than ten percent of our power grid. This thrust Texas into a situation in which outages occurred."

Except that's horse manure. Indeed, earlier, in a setting (local news)  different from the FOX Liars Axis, Abbott had come clean and told the truth, e.g.  "The companies that generate the power, their operations have frozen up and trip wired, and become non-operational. This is why there is a lack of power, a lack of natural gas arriving at power generation centers across the state.  That's because the ability to both manufacture and transport the natural gas has been frozen also. It's been frozen in the pipeline, frozen in the rig,  frozen in the transmission line."   

So  which account do you believe?  The one given to the local news outlet or to the Liars Axis on FOX?  The latter being the one where Tucker Carlson has also been spinning lies about the windmills being responsible because "when green energy came to Texas it became totally reliant on windmills.  It got cold and the windmills broke.  That's what happens in the green new deal.  You're without power. Millions are still without power."  

As Chris Hayes sarcastically put it in response on ALL In last night: 

"Yes, unbeknownst to most people, Texas became some sort of hippie commune, relying on windmills, and rainbows and flower power. Totally reliant! He actually said that. Did Alexandria Ocasio Cortex ten years ago, at the tender age of 21,  parachute into the state of Texas and put up windmills under the cover of night?  Were she and Al Gore just driving through the Permian Basin secretly unplugging all the rigs? How stupid does Tucker think his audience is exactly?"

  This is the crazy blame game the Right's nuts and liars are playing while Texans are freezing their butts off-  or nearly so- in their own homes. 

The main immediate problem was frigid temperatures that stalled natural gas production, which is responsible for the majority of Texas’ power supply.   Wind makes up just a fraction — 7 percent or so, by some estimates — of the state’s overall mix of power generation.   But just like Minnesota, Texas could have insulated its wind turbines as opposed to letting them be exposed to the elements, just like it could have winterized its natural gas pipelines.  But it chose not to in order to keep costs down, and thus bet the house nothing terrible would happen.  It lost.

The way the state’s wholesale power market works, utilities have very few incentives for investment in backup power or protecting energy infrastructure. The state does not say to a generator, “Please build us some extra backup power plants, and we will charge that cost to our customers.”  Why the hell would they if the main point of a deregulated energy market is to keep costs as low as possible?   So you don't spend good money on investments that are costly and might never be needed anyway.  But... if they do come, well, you reap the bloody whirlwind - as Texas has done.

 Perversely, at the same time, the market allows a power generator to charge excessively high prices when available supply falls short — which, for an investor, could be a long shot. As a result, ERCOT’s backup power, called the spinning reserve, is lower than most other areas in the United States.  But when that reserve has to be tapped -  like in this week's frigid spell-  Katy bar the door, because consumers will have to pay through their asses.  (The way Texas wholesale market works, backup plants charged an eye-popping $9,000/mwh rate this week - while the price was $30/mwh just six days ago).

That is why some people in Houston -  such as shown on ALL In last night- were getting bills for $1,622 and up.   Incredibly, former TX governor Rick Perry proclaimed that Texans would rather go three days without heat than let the feds tell them what to do. Well, we'll soon see about that, eh?

How did this happen?  While ERCOT had anticipated rolling blackouts and warned customers to reduce power use and expect brief outages, the demand for power far exceeded the available supply, leaving millions stuck in the dark and the cold.  The spinning reserve had to be tapped, and customers dinged with super high bills.

Texas simply doesn't have a large enough backup system for when power demand shoots way up, or when  regular generators go offline, as they did this week. It is a problem that plagues ERCOT every year as the hottest part of the summer approaches. This is the fault of Texas' deregulated, cheapo, wholesale market structure.   Even after a winter storm in 2011 caused widespread power outages a report recommended that Texas winterize its generators, including doing what MN does for its wind turbines: using heating packs and insulated pipes. Texas declined to address the recommendations. Too costly!  The rest, shall we say, is history.

 Want things to improve?  Either ditch the wholesale market structure  in favor of one incorporating backup plans  and power sharing (with other grids) or abandon the independent  grid idea and join the eastern or western connection. (Like El Paso has)


See Also:

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 The Texas grid got crushed because its operators didn’t see the need to prepare for cold weather

AND:

The latest GOP nonsense on Texas shows us the future Republicans want


Evaluating The Dilogarithmic Function

Among the many fascinating mathematical curiosities is the so-called dilogarithmic function. It was believed to be originally discussed by Leonard Euler in 1729 in the form:

Li2 (x) = å ¥n=1   [xn  / n2]                            


Whereby  Euler was unable to evaluate Li2 (1)   from the integral form. However, Landen, in a paper appearing in Philosophical Transactions (Landen, John: 1760,  ‘A New Method of Computing The Sums of Certain Series, Vol. 51, Pt. 2, p. 553)showed the evaluation required the use of log (-1). Unaware of Euler’s work, Landen set out to determine log (-1) by first differentiating x =   sin (z) to obtain: 

dz/ i=   dx/ [x2 -1]1/2

Integrating this, he got:

z/ i = log (x + [x2 -1]1/2  / i

where i= [-1]1/2 

Landen then set z = π/2 and x = 1 so that log ([-1]1/2) = log(i) = -π/2i

Since a square root must have two values, he concluded: log (-1) = +/-  π/ i

Then to evaluate Li2 (1) he commenced with the series form:


x-1 + x-2 /2 + x -3/ 3 + ……= log (1/ 1 – x) = log x + log (1/ 1- x) - π/i


Landen then divided the preceding by x to obtain:


- Li2 (1/x) = - π/i log x + ½ (log x)2 + Li2 (x) + C



To find C, Landen took x = 1, and then using algebra (solving for C above) obtained:

C = -2 å ¥ n=1  (1/n2)


He then had to evaluate the series above and did this by setting x = -1 in the expression for - Li2 (1/x) . Using a (+) sign this time, he arrived at:



å ¥ n=1  (1/n2) = π2 / 6

Landen’s computations indicated that the formula for - Li2 (1/x) was only correct for the case x > 1. His tactic of dividing by x, then integrating led ultimately to the recognition that the series Li2 (x) and Li2 (1/x) converged when x = exp(i Θ) for the interval:

0 < Θ < 2π

Note, however, that if this is substituted directly into the expression for - Li2 (1/x) the result is invalid. However, choosing the (+) sign instead makes it valid, yielding the Fourier series expansion of the 2nd Bernouli polynomial.