Monday, June 23, 2025

Solutions To Complex Numbers (Polar Form) - Part 1

 1)1) Express each of the following end results in the form: r exp(iq):

a) (2 + 3i)(1 – 2i)

b) (1 + i) (1- i)

c) (1 + Ö-3)2

Solutions:

a) Multiplying the two factors we get:

2 + 3i -6(i2) -4(i) = 8 + 3i – 4i = 8 –i

Next convert to standard polar form, given :

z = x + iy = 8 - i

so: r = r = [82 + (-1)2]1/2 = Ö65  =   8.1

q = arg(z) = arctan (y/x) = arctan (-1/8) = -7.1 deg.

So: z = 8.1 [cos(-7.1) + isin(-7.1)] 

     Or: z = 8.1 exp (i(-7.1))



b) 
(1 + i) (1- i)

Expand to get: 1 + 1i - 1i -(i)(i)  = 1 - (-1) =  1 + 1 = 2

For which arg(z) = 0  (No imaginary component, i)

And arctan (0) = 0 hence there is no polar form.

But: z =  r exp (i 0) = Ö2 (1) = Ö2

c) (1 + Ö-3)2

Expand to get:

z = (1 + Ö-3) (1 + Ö-3)

But bear in mind that: (Ö-3) = [iÖ3)] so we have:

(1 + [iÖ3]) 
• (1 + [i Ö3]) = 1 + 2i[Ö3)] + (-1)3 = -2 + 2i[Ö3)]


z = -2 + 3.46i

r = [(-2)2 + (2(iÖ3) 2] =  [4 + 36]1/2 = [40]1/2 = 6.3


q = arctan (y/x) = arctan (2[Ö3)]/ -2) = arctan (-Ö3) = (-60) = -p/3
Thus, z = 6.3 exp(i(-p/3))

2) Plot the results of (b) and (c) on the same Argand Diagram and obtain the resultant.


Solution: 

We plot the results of (b) and (c) on the Argand diagram and obtain the resultant

Call z(b) = 0 + 
Ö2 i = 1.4i (from that solution, i.e. z =  r exp (i 0) )

And z(c) = -2 + 3.46i

The plots of the two numbers are shown in the diagram. The resultant is sketched and can be confirmed from the algebra:

z(b) + z(c)= 1.4i + (-2 +3.46i) = -2 + 4.86i



This yields a parallelogram with points at: (0, 0), (0, 2), (-2, 4.86) and (-2, 3.46). 



Friday, June 20, 2025

An Extended Explanation of The Summer Solstice

 

The seasons in perspective for Earth's orbit
                                                         Focus on summer solstice

Today marks  the first day of summer, when the Earth reaches the summer solstice in its orbit.  So we are talking about not only a specific time (8:42 p.m, Mountain time today for Colorad0) but also a position and orientation in space. That position and orientation is what typically makes the days just before and after the summer solstice  warmer  in the Earth's northern hemisphere. (Our days here in Colorado have suddenly soared into the mid 90s.)

And note this has nothing to do with the distance from the Sun. In fact, on today's date Earth is actually approximately 1.5 million miles further away from the Sun, than it would be at the winter solstice (December 21). The diagram below clarifies this in terms of Earth's elliptical orbit:

Here, the generic (standard) date (June 21) for the summer solstice - as a position on the orbit  - is approaching the aphelion or greatest distance from the Sun. (Occurring July 3rd). Conversely, we see the winter solstice position in the orbit approaching the perihelion (nearest distance from the Sun) - which is on January 4th. The point is that distance from the Sun is not the determinative factor in the heating of the  northern hemisphere, but it's rather the inclination or tilt of the toward the Sun (seen in better relief in the 2nd diagram from the top.)

We can now go through the procession of one season to the next using the top illustration - which of course is not to scale.

Note first,  point A in the diagram (far left) , we have the commencement of summer in the northern hemisphere (summer solstice) because the Earth's axis is tilted toward the Sun hence the N. hemisphere receives more direct sunlight, radiation. Also at point A, there is the longest day.  This date is given the name "summer solstice". 

What makes it so?  Examining the diagram carefully you can see the greatest fraction of sunlit surface appears at this date.  (Adjacent dates, i.e. June 20, and June 22, approach this but don't equal the same large fraction.).  This is given the Earth's axis is now fully tilted (at 23.5 degrees) toward the Sun.  This greatest fraction of sunlight translates into the greatest duration of daylight for the date. Hence the longest day.  

Prior to the summer solstice and extending all the way from the date of the winter solstice (on or about Dec. 21st, at extreme right) the days are continually increasing in length as the fraction of sunlit surface continually increases. Note also at each of the equinoxes (March 21st - Spring or Vernal ;  Sept. 23rd- Autumnal) the lengths of the day are the same, at 12 hrs.   Hence the term "equinox" or "equal night". 

 In summary then, the tilt of the Earth modulates the extent or the amount of sunlight-radiation (including heat) a hemisphere receives at a particular time, or over a given interval. Onset of winter and summer are defined respectively by the extreme axial tilts of  either "fully away" from the Sun, and "fully toward", while Fall, Spring are defined by neutral tilts - i.e.  neither away nor directly toward.

Finally, it is important to point out that the "Longest day" on June 21st applies to the northern hemisphere only. For the southern hemisphere it is the shortest day while our winter solstice is the longest.  Again,  this can be grasped by examining the diagram and noting while the proportion of sunlit surface is large for us (in the N. Hemisphere) it is receding for the south.  The converse is true when one  looks to the image for Earth on the opposite side (Dec 21).

See Also:




And:

Thursday, June 19, 2025

Wall Street Journal Weekend Piece Presents Winners & Losers In "Big, Beautiful Bill" Boondoggle

   Graph showing winners and losers (source: Congressional Budget Office)
 

"When autocrats, plutocrats and their lobbyists converge on the House and Senate there is really only one thing to expect - the rich get richer and the poor get the shaft.

It's the GOP doctrine that the only reason that people are poor is that they don't try hard enough, and so do not deserve help.

The really amazing thing is that the people they short-change the most are the most ardent supporters of the people who treat then so badly."  - WaPo Comment


The Wall Stret Journal weekend column (p. A2) headed: 

'CBO Says Megabill Would Help Rich And Hurt The Poor

Did us a favor by breaking down all the stats needed to see how this lollapalooza would screw over all those who can least afford it, including multitudes of MAGAs.  As The Journal notes:

"The study by Congress' nonpartisan scorekeeper looks at the combined effects of many of the House bill's tax reductions and it cuts to federal programs including Medicaid and nutrition assistance."

Adding (See graph):

"The bill's benefits increase steadily with income. Households near the middle of the income distribution would see an increased benefit of $500 to $1,000 as an annual average.  The top 10% of households would see a gain of about $12,000 or 2.3% of income, largely because of tax cuts. (This is compared to 0.5%- 0.8% of income for the middle.)

The bottom 10 % of households would come out worst off, losing about $1,600 a year - mainly because of benefit cuts."

Meanwhile this same edition of the bill - if unmodified by the Senate - would "permanently extend Trump's 2017 tax cuts" - effectively adding $3.3 trillion to the national debt, e.g.

Trump’s bill could add $3.3 trillion to national debt, CBO finds

This means, if passed, the interest on the debt could double from $900 billion to $1.8 trillion by 2034. Your children and grandchildren will be coughing up nearly half their salaries on paying the interest on this debt, meaning vastly less affordability of everything across the board - from food to fuel to medical.

 In a WSJ follow-up piece yesterday ('GOP Plan Hikes Deficits', p. A4) there was presented one of the most salient observations:  

"In all, the bill would increase real gross domestic product by 0.5% over the decade.  That is far short of the 2.8%  annual growth rate House Republicans claim. The additional budget deficits would counteract that growth, crowding out private investment and driving up interest rates by an average of 14 basis points above CBO forecasts.  The bill would then add to budget deficits by $2.4 trillion through 2034 - on top of the $21 trillion the country would incur if Congress does nothing  and the 2017 tax cuts expire as scheduled at the end of the year."

To fix ideas, it is instructive to see the effects of this monstrosity on my own state of Colorado.  Basically, it could push food assistance in Colorado into a vicious cycle of funding cuts.  In addition, a peril of increasing mistakes would emerge in determining eligibility, i.e. for SNAP and Medicaid,  and further funding cuts to punish those mistakes, Gov. Jared Polis warned in a letter to congressional leaders 

The vile bill which passed the House and is currently in the Senate, would require states to pay 75% of the cost to administer the Supplemental Nutrition Assistance Program, or SNAP, which offers low-income people a monthly allowance to purchase food.

Currently, the federal government splits the cost evenly with states. If the ratio shifted, Colorado would have to come up with an additional $47 million to fund the workers at county human services departments who determine SNAP eligibility.

The bill, backed by President Donald Trump, would also require states to start paying a share of the food benefits themselves. The state estimates it could have to come up with anywhere from $130 million to $259 million annually to pay its share, depending on the language in the final bill.

Colorado can’t come up with that amount of cash, particularly given the state’s constitutional limits on raising taxes, Polis — a Democrat — and 10 people working in human services and agriculture said in the letter. State lawmakers had to close a $1.2 billion budget hole for the upcoming fiscal year, and warned that Colorado could face an even bigger gap next year, with fewer options to fill it.

“Devoid of purpose, this provision is simply a budget gimmick in Washington, D.C., and a devastating blow to states,” the letter said.

If funding for counties doing the work decreases, mistakes will increase, the letter said. Since the bill includes a provision punishing states with higher error rates with more funding cuts, the damage could spiral, particularly since the bill doesn’t include any time for the states to prepare, the letter said. If that happened, the state’s share of the benefits could rise as high as $324 million.

Colorado previously lost about $13.1 million in funding from the U.S. Department of Agriculture to help schools and food banks purchase locally grown produce. The USDA also reduced emergency food assistance because of concerns that undocumented immigrants benefited from it.  

Overall, 42 percent of Americans oppose the budget bill “changing tax, spending and Medicaid policies” that narrowly passed in May by the Republican-controlled U.S. House of Representatives, compared with 23 percent of Americans who support the bill, and 34 percent who say they have no opinion.

Opinions on extending Trump’s 2017 tax cuts are mixed. A 71 percent majority supports extending tax cuts for individuals with incomes under $100,000, as do 53 percent when asked about individuals with incomes between $100,000 and $200,000. But support drops to 29 percent for lengthening tax cuts for individuals with income above $400,000 and 30 percent for corporations.

In the end, the GOP seems not to grasp that they are literally 'swimming against the public tide' by pushing this 1.000 page offal on us. The fact is Americans broadly support increasing or maintaining existing levels of funding for popular safety net programs including SNAP, Medicaid and Social Security and Medicare. This according to the Associated Press-NORC Center for Public Affairs Research. Here are some aspects of opposition to the Reeps policy agenda the AP/NORC study found:

- Most Americans want Medicaid and SNAP funding increased or kept as it is.

- Many, indeed, see Medicaid and food assistance programs as underfunded even as the GOP Congress proposes significant cuts to Medicaid and food & nutrition assistance.

- About half of U.S. adults say "too little" funding goes to Medicaid, which is a government health care program for low income people and those with disabilities.

- 45% say food and nutrition program funding needs to stay the same or be increased.

Lastly, as recent AARP Bulletin columns have observed, there is nearly $350 billion in UNPAID care already in this country. That covers single (often older) caretakers of seriously ill or disabled relatives on Medicare or Medicaid.  Thus, they should not be forced into working for their benefits as the GOP's scrooges have proposed. 

See Also:

Trump’s Medicaid cuts could break the MAGA coalition


Excerpt:

As Republicans crafted their One Big Beautiful Bill, Sen. Josh Hawley (R-Missouri) offered a warning: “Our voters support social insurance programs. More than that, our voters depend on those programs.”

While House Republicans ignored his pleas, opting to pass a bill with cuts to Medicaid that the Congressional Budget Office estimates would lead to 7.8 million fewer people with coverage, Hawley was onto something. There’s strong evidence that voters whom President Donald Trump brought into the Republican Party over his three presidential runs will be especially affected by the proposed funding changes to health care.

A quick glance at the counties with the biggest Republican swings in the Trump era bears this out: In county after county and state after state, Trump made disproportionate gains among lower-income voters, many of whom rely on Medicaid — a program that covers nearly 80 million Americans nationwide.


And:


Excerpt:

If you don’t have anything nice to say, just make something up instead.

That’s the strategy Republican politicians have adopted in hopes of selling their regressive, unpopular budget bill to voters, as an even harsher version of the legislation now makes its way through the Senate.

The One Big Beautiful Bill is already deeply underwater, voter-wise. Multiple polls now show that about twice as many Americans oppose the bill as support it. This should not be a surprise, given that this legislation is effectively a mash-up of multiple past GOP initiatives that, individually, had each been among the worst-polling major bills in recent history.

For example, the OBBB includes tax cuts that disproportionately benefit the wealthy; when Republicans last did this, with their 2017 tax overhaul, the legislation garnered more opposition than support in virtually every poll. The OBBB also includes a sort of backdoor Obamacare repeal by slashing Medicaid and other measures. As battle-scarred Republicans might remember, their last attempt to repeal the Affordable Care Act so enraged voters that the party suffered a watershed wipeout in the subsequent midterms.  

Here, voters’ revulsion happens to be grounded in data (which, candidly, is not always the case). As I’ve noted before, the OBBB could easily become the largest single transfer of wealth from poor to rich in U.S. history. 


And:



Excerpt:

H.R. 1, the One Big Beautiful Bill, manages to be the worst of both worlds. It spends an eye-watering amount our country can ill afford and directs almost all of that money to the wealthiest while cutting critical services for low- and middle-income Americans.

The bill’s significant cuts to health care and food programs have drawn the most attention. But even a tax provision included to help families, by increasing the child tax credit, would block the neediest households from receiving new benefits while piling on more aid for those with higher incomes.

We study poverty, taxes and the safety net, and we have researched the child tax credit for years. Under current policy, the credit promises families up to $2,000 per child annually. In reality, 1 in 4 children nationwide cannot receive this full amount.

The reason: The credit requires a minimum income level to qualify at all and then slowly scales up with family income until hitting the maximum amount. This “trapezoid” structure leaves children in low- and moderate-income families ineligible for the full credit. H.R. 1 not only keeps this structure but also ensures that many households with two working parents would receive no additional help from the expanded credit.

H.R.1 would raise the maximum credit to $2,500 per child. A two-adult, two-child family could, in theory, gain up to $1,000. But because the bill doesn’t change the credit’s formula, it would make it harder for families to access the higher credit. The same size family would need an additional $12,000 in income, or at least $48,000 total, to receive the full $5,000 credit for their two children ($2,500 each).

This effectively locks out families with incomes below $60,000 of any meaningful gains. Nationwide, 22 million children — 1 in 3 — would be ineligible for the full $2,500 credit. The majority of those children, 17 million, would see nothing at all from the credit increase, while 5 million would see only a partial increase. That means families with higher incomes, up to $400,000, would reap the full benefit, while families just scraping by would not.


And:


Excerpt:

It is hard to think of a less coherent pair of policies: President Trump’s tax policy encourages the very offshoring that his tariffs are intended to stop.

Take the more than $600 billion pharmaceutical industry. Over the past few months, Mr. Trump and his associates have repeatedly criticized companies’ moves to offshore much drug making, particularly to Ireland. “We can’t be beholden and rely upon foreign countries for fundamental things that we need,” Commerce Secretary Howard Lutnick said on April 13.

But the tax incentives in Mr. Trump’s 2017 tax and spending package helped generate this problem in the first place — a problem that would continue under the Republican bill under consideration. Mr. Trump’s Tax Cuts and Jobs Act created a loophole that made it far more profitable for the pharmaceutical giants, including Eli Lilly, Pfizer, Johnson & Johnson and Merck, to manufacture some of their most profitable drugs in Ireland. Unsurprisingly, that is what happened, with America’s imports of pharmaceuticals soaring to $250 billion in 2024, way up from $110 billion in 2016.

And:

by Imran Khalid | June 18, 2025 - 4:54am | permalink

— from Foreign Policy In Focus

In the shadow of a Trump belligerent presidency and with an economy staggering under its own contradictions, the United States finds itself teetering on a precarious precipice where populist policymaking, fiscal illusions, and dwindling international faith in U.S. Treasury bonds are aligning to form a slow-motion crisis. Whether it’s labeled “Big Beautiful Bill” or cloaked in the language of patriotic economic revival, the endgame looks less like American greatness and more like an exercise in fiscal self-sabotage.

As of early June 2025, the U.S. Treasury yield curve—often a harbinger of economic angst—has begun blinking bright red again. Goldman Sachs, in its latest report, now predicts 30-year Treasury yields could exceed 6 percent, with 10-year yields potentially climbing above 5 percent, levels unseen in more than two decades. The spike is no statistical anomaly. It’s symptomatic of a deeper malaise: a confluence of monetary hardening, fiscal recklessness, and eroding global trust.

» article continues...

And:

Senate Republicans propose deeper Medicaid cuts. Here’s what that means.


Excerpt:

Senate Republicans are preparing to slice deeper into Medicaid to finance President Donald Trump’s agenda, proposing legislation that analysts say could deal a greater financial blow to hospitals and result in millions more uninsured Americans.

The measure reflects Republicans’ willingness to cut the nation’s safety-net health insurance program, despite qualms expressed by some moderates and populists within their ranks.

“The Senate just made a bad bill worse,” Chip Kahn, CEO of the Federation of American Hospitals, said in a statement. “Rural communities would take the hardest hit, with struggling hospitals compelled to face difficult decisions about what services to cut.

And:

This tax testifies to the utter corruption of conservative thought

Excerpt:

 There’s so much wrong with the sprawling budget bill that the House just passed and the Senate is debating — I refuse to call it by its cringey name — that it seems myopic to fixate on a provision that probably wouldn’t make anyone’s top 10. And yet I find myself stuck on this new tax Congress would impose on electric cars. Not because of the policy itself, but because the way it’s being enacted testifies to the utter corruption of conservative thought.

Technically speaking, the proposed $250 annual levy on EVs (and $100 annually for owners of hybrids) isn’t a tax — it’s a fee. 


Wednesday, June 18, 2025

Can Authoritarianism Migrate Into Dreams & Generate Nightmares? Yes, According to German Journalist

  

                                "I'm  comin' for your dreams, babe!"


       We need to be careful not to become obsessed with Trump or his authoritarian streak. Why not?  Because it can result in nightmares with him camped out in our dreams-nightmares - not just outer reality. So argues Charlotte Beradt in her Third Reich Of Dreams, which documents over a 6-year period how the Nazis and Hitler's dictatorship slowly colonized the unconscious of millions of Germans.  In effect, the Third Reich's persecutions and propaganda had seeped into the last refuge of the private self: dreams.

These unsettling findings in respect of Beradt's work were recently the focus of a WSJ book review, 'Nightmares Around The Clock', May 17-18, p. C12, by Benjami Balint.  He indeed opens by quoting her own words after writing: 

"One night in 1933, after Hitler had taken power, a Jewish journalist in Berlin awoke from uneasy dreams, drenched in sweat. 'The thought occurred to me', she later wrote,

'that I might not be the only one among thousands and thousands to be condemned to such dreams by the dictatorship. The things that filled my dreams must fill theirs too - breathless flight across fields, hiding at the top of towers dizzying height, cowering down below in graves - everywhere the Storm troopers at my heels."

Heady stuff, but is it real, as in real dreams?  The WSJ reviewer clearly believes they were, noting: "By the time she fled Germany in 1939, Beradt had smuggled out a unique archive of inner life under tyranny.  After the war, in New York, she shaped these oneiric fragments into a potent volume, 'The Third Reich of Dreams'.:

Mr. Balint is also careful to point out the dreams recorded "are not symptoms of early neurosis or early childhood experiences."   They were, in fact, "revelatory account of despotism internalized."

I can believe it because my late German friend Kurt Braun - who had been forced into the Hitler Youth at age ten - recalled years of such dreams after the war. Mostly being lined up for the Fuhrer's inspection, and then not saying 'Heil Hitler!' fast enough.

                              Kurt showing us archival film of Nazi terrors in July, 1978

As he told us on our first visit to his home in Frankfurt-Am-Mein in 1978:

"The dreams, usually nightmares, arrived almost every night. Finally, I realized on becoming a grown man the only way to fend off these night demons was to gather archival films of what the Nazis actually did."

The strategy worked, as he essentially banished the Hitler dictatorship occupation from his unconscious, transferring them to the conscious mind via films. (He showed us several of them, and they were not for the squeamish.)

When one reads the WSJ review, and in particular the first sentence from the paragraph near the end of the third column: "Several Germans described to Beradt dreams in which dreaming itself had been outlawed."

One wonders if this wasn't also Kurt's experience and what really drove him to collect dozens of archival reels of the time.  Well, look at it this way, his dreams might have been "outlawed"- but not watching the archival films in objective reality.

Balint's words at the end of his review are worth considering:

"Her book's involuntary testimonies are dictated, so to speak, by a particular dictatorship. But in our own moment of renewed anxiety about authoritarian impulses worldwide, they make us wonder:

What happens when the dreams are no longer fully our own? The leastwe can do is read this singular document with our eyes open."

See Also:

by Jeffrey C. Isaac | June 14, 2025 - 4:45am | permalink

The Trump administration has proceeded, and is proceeding, at a furious pace to employ executive power attack and undermine central pillars of American democracy, rendering a system already threadbare and fragile even more weakened.

This past week’s deployment of National Guard troops, and then U.S. Marines, on the streets of Los Angeles, in defiance of California Governor Gavin Newsome and LA Mayor Karen Bass, is but the latest example.

Tomorrow—Saturday, June 14—the administration is planning an enormous military parade on the streets of Washington, D.C., to celebrate “Flag Day,” the U.S. Army, and especially to celebrate Trump and his birthday.

» article continues...

And:

by Heather Digby Parton | June 15, 2025 - 5:25am | permalink

— from Salon

In March 2024, Donald Trump hosted Hungarian Prime Minister Viktor Orbán at Mar-a-Lago. After years of receiving praise and pilgrimages from the American right, the autocrat had begun appearing at conservative events in the U.S. Instead of visiting the White House to see then-President Joe Biden, Orbán spoke at the Heritage Foundation — the conservative think tank behind Project 2025 — before heading to Florida. “He’s a non-controversial figure," Trump told a crowd at Mar-a-Lago upon Orbán's arrival, "because he says, ‘This is the way it’s going to be,’ and that’s the end of it. Right? He’s the boss.”

Trump loves that.

Orbán's Hungarian regime has often been characterized as modern authoritarianism or a "hybrid regime of electoral autocracy," one in which power is accumulated by the ruling party over time through creative quasi-legal means. He focused on transforming institutions by changing election laws, allowing him to create legislative supermajorities while winning a mere plurality of the vote. He packed the courts with loyalists, found friendly oligarchs to buy up independent media and took over universities — or forced them to close down. He hammered on culture war issues of immigration, nationalism and family values.

» article continues...

And:

The Insidious Danger Of Propaganda And How It Has Infected Brains And Threatened Our Democracy

And:

Archival Photos Remind Us That Book Burning Launched Third Reich's Savagery In 1930s

And:

Original Sin? NOT Dems 'Covering Up' Biden's Age Faults - But Voters Putting a Fascist Into Power For the 1st Time In U.S. History


Solving More Difficult Partial Differential Equations (Pt. 2)

  In line with the topic of difficult partial differential equations, we consider the problem of the electrical potential distribution for an insulated cylinder with cross-section shown below:


A partial differential equation can be applied starting with two particular forms of what is called the Poisson equation.    For the case with position vector r  such as shown for the cross section shown, we write:

Ñ 2  j  =   f(r)

For the case  f(r) = 0 we obtain the Laplace equation:

Ñ 2  j  =  0  

Based on the proceeding the equation must be satisfied in a charge free region 

j (r)=  0 

The field E can easily be found from the electrostatic potential j which must satisfy the Laplace equation written in cylindrical coordinates.   

From symmetry considerations  j must be independent of z so that: j = j (r, q)  and:

1/r  / r (r   j /¶ r)  +   1/ r 2  (  2 j / q2 )  =  0 

 The following boundary conditions are imposed on j:

j (r, q)  =   +V  (0  <  q <  p )

j (r, q)  =   -V  (- p  <   q  < 0)


These basics allow us to proceed with the solution which can be applied to analogous PDEs. In this case it is understood that  j must be periodic everywhere inside the cylinder (r < a).  But it is expected that this condition will break down at r = a,   q = 0 or p  (where the insulation is) since E is expected to be infinite at these points.  The method of separation of variables can be used again such that:

j (r, q)  = R(r) q(q)


Then:

1/r  d/dr[r (dR/dr) Q ]+   1/r 2  R(r) d 2 Q/d q 2  =  0

Þ     

1/r  [dR/dr Q + r d 2R/dr 2  Q ]+  R/r 2  d 2 Q/d q 2  =  0

 

Þ        r  dR/dr + r 2 d 2R/dr 2  = -R/q   d 2 Q/d q 2 

Þ     r 2  /R {d 2 R/dr2 } +  r/R  dR/dr = - 1/q d 2 Q /d q 2 


We next let l  be the separation constant, yielding.

1) r 2   d 2R/dr 2  + r  dR/dr + lR =  0

2) d 2 Q /d q 2 =  lQ

Leading to:

(3) 1/r  d/dr[r (dR/dr) Q ]+   l/r 2  R =  0

Since  Q  is periodic in q  then  Q must be also which immediately implies that: l = - m 2

(m = 0, 1, 2, 3,....)

And Q is then a linear combination of sin m q, and  cos m q 

The radial equation (3) becomes:

 d 2R/dr 2  +  1/R  dR/dr  -  m 2 /r 2  R =  0

which is just the Euler equation with the solutions r m  and r -m

for m > 0,  C = const.  and log r (for m = 0)  

Since  j (r, q)  must have a gradient at r = 0,  the solutions   
  r -m and log r are not admissible.  According to the applicable general principles we can form a linear (infinite) superposition of solutions, i.e.

j (r, q) =   
A o  +   å¥ m = 1   r -m (A m  cos mq   +    B m  sin mq ) = f(q)
 
To satisfy the boundary conditions we set:

 j (r, q) = f(q)  =  {+V  (0  <  q <  p )

                              { -V  (- p  <   q  < 0)  

So the constants A m   and B m  are related to the Fourier coefficients of f(q).  Then A o  = A m  (all m) owing to the asymmetry of  under the interchange  q -> - q.  Note that this property could have been inferred for j (r, q)   as well.  Being aware of these facts can considerably simplify solutions in more complicated PDE problems.  In this case direct calculation leads to the result:

B m  = 2V/p a m   ò p  0  sin mq dq   = {0  (m even)

                                                                 { 4V/ a m p m     m odd


So that: j (r, q)  = 4V/p  å¥ m = 1,3,5...   1/m (r/a) m sin mq

Note: It's not difficult to verify that the above function satisfies the conditions of the problem.   I.e. if r < r1  , where  r1 is any positive integer less than a, the series can be differentiated term by term twice with respect to both r and q  .  The factor (r/a) will lead to uniform convergence in the 2x differentiated series. Thus the 'difficult' PDE is easily verified. Try it and see!

   Challenge Problem:

 A partial differential equation:    

  u/  t   =   c 2   [ 2 u/  x2 ]

with u = u(x, t)

Such that 0 <  x <  ℓ   and t > 0

is used when considering a perfectly insulated rod of length (with no lateral heat loss) We include the initial and end conditions for the rod:  u(0, t) = u(ℓ  , t)    

And: u (x, 0) = f(x)

Where u represents the temperature of the rod as a function of position x.  Obtain the solutions in terms of Fourier series for:

1) u( x, 0) = f(x)

and:

2) u(x, t) =  u(ℓ, t) 

Hint: Use a variables separable method such that:

u(x, t) = X(x)  T (t)

So that:   2 u/  x2  =  X''  T

And:   X T' =  =    c 2  X''  T