# The Lumber Room

"Consign them to dust and damp by way of preserving them"

## Stephen Fry’s “The Ode Less Travelled”: Foreword

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I reproduce below the line below Stephen Fry’s entire foreword to his book The Ode Less Travelled, because I find myself frequently referring to it and would like to be able to direct friends to some place to read it‌ — this is now such a place.

I HAVE A DARK AND DREADFUL SECRET. I write poetry. This is an embarrassing confession for an adult to make. In their idle hours Winston Churchill and Noël Coward‎ painted. For fun and relaxation Albert Einstein played the violin. Hemingway hunted, Agatha Christie gardened, James Joyce sang arias and Nabokov chased butterflies. But poetry?

I have a friend who drums in the attic, another who has been building a boat for years. An actor I know is prouder of the reproduction eighteenth-century duelling pistols he makes in a small workshop than he is of his knighthood. Britain is a nation of hobbyists—eccentric amateurs, talented part-timers, Pooterish potterers and dedicated autodidacts in every field of human endeavour. But poetry?

An adolescent girl may write poetry, so long as it is securely locked up in her pink leatherette five-year diary. Suburban professionals are permitted to enter jolly pastiche competitions in the Spectator and New Statesman. At a pinch, a young man may be allowed to write a verse or two of dirty doggerel and leave it on a post-it note stuck to the fridge when he has forgotten to buy a Valentine card. But that’s it. Any more forays into the world of Poesy and you release the beast that lurks within every British breast—and the name of the beast is Embarrassment.

And yet…

I believe poetry is a primal impulse within us all. I believe we are all capable of it and furthermore that a small, often ignored corner of us positively yearns to try it. I believe our poetic impulse is blocked by the false belief that poetry might on the one hand be academic and technical and on the other formless and random. It seems to many that while there is a clear road to learning music, gardening or watercolours, poetry lies in inaccessible marshland: no pathways, no signposts, just the skeletons of long-dead poets poking through the bog and the unedifying sight of living ones floundering about in apparent confusion and mutual enmity. Behind it all, the dread memory of classrooms swollen into resentful silence while the English teacher invites us to ‘respond’ to a poem.

For me the private act of writing poetry is songwriting, confessional, diary-keeping, speculation, problem-solving, storytelling, therapy, anger management, craftsmanship, relaxation, concentration and spiritual adventure all in one inexpensive package.

Suppose I want to paint but seem to have no obvious talent. Never mind: there are artist supply shops selling paints, papers, pastels, charcoals and crayons. There are ‘How To’ books everywhere. Simple lessons in the rules of proportion and guides to composition and colourmixing can make up for my lack of natural ability and provide painless technical grounding. I am helped by grids and outlines, pantographs and tracing paper; precise instructions guide me in how to prepare a canvas, prime it with paint and wash it into an instant watercolour sky. There are instructional videos available; I can even find channels on cable and satellite television showing gentle hippies painting lakes, carving pine trees with palette knives and dotting them with impasto snow. Mahlsticks, sable, hogs-hair, turpentine and linseed. Viridian, umber, ochre and carmine. Perspective, chiaroscuro, sfumato, grisaille, tondo and morbidezza. Reserved modes and materials. The tools of the trade. A new jargon to learn. A whole initiation into technique, form and style.

Suppose I want to play music but seem to have no obvious talent. Never mind: there are music shops selling instruments, tuning forks, metronomes and ‘How To’ books by the score. And scores by the score. Instructional videos abound. I can buy digital keyboards linked to programmes that plug into my computer and guide me through the rudiments, monitoring my progress and accuracy. I start with scales and move on to chords and arpeggios. There are horsehair, rosin and catgut, reeds, plectrums and mouthpieces. There are diminished sevenths, augmented fifths, relative minors, trills and accidentals. There are riffs and figures, licks and vamps. Sonata, adagio, crescendo, scherzo and twelve-bar blues. Reserved modes and materials. The tools of the trade. A new jargon to learn. A whole initiation into technique, form and style.

To help us further there are evening classes, clubs and groups. Pack up your easel and palette and go into the countryside with a party of like-minded enthusiasts. Sit down with a friend and learn a new chord on the guitar. Join a band. Turn your watercolour view of Lake Windermere into a tablemat or T-shirt. Burn your version of ‘Stairway to Heaven’ onto a CD and alarm your friends.

None of these adventures into technique and proficiency will necessarily turn you into a genius or even a proficient craftsman. Your view of Snow on York Minster, whether languishing in the loft or forming the basis of this year’s Christmas card doesn’t make you Turner, Constable or Monet. Your version of ‘Fur Elise’ on electric piano might not threaten Alfred Brendel, your trumpet blast of ‘Basin Street Blues’ could be so far from Satchmo that it hurts and your take on ‘Lela’ may well stand as an eternal reproach to all those with ears to hear. You may not sell a single picture, be invited even once to deputise for the church organist when she goes down with shingles or have any luck at all when you try out for the local Bay City Rollers tribute band. You are neither Great Artist, sessions professional, illustrator or admired amateur.

So what? You are someone who paints a bit, scratches around on the keyboard for fun, gets a kick out of learning a tune or discovering a new way of rendering the face of your beloved in charcoal. You have another life, you have family, work and friends but this is a hobby, a pastime, FUN. Do you give up the Sunday kick-around because you’ll never be Thierry Henry? Of course not. That would be pathologically vain. We don’t stop talking about how the world might be better just because we have no chance of making it to Prime Minister. We are all politicians. We are all artists. In an open society everything the mind and hands can achieve is our birthright. It is up to us to claim it.

And you know, you might be the real thing, or someone with the potential to give as much pleasure to others as you derive yourself. But how you will ever know if you don’t try?

As the above is true of painting and music, so it is true of cookery and photography and gardening and interior decoration and chess and poker and skiing and sailing and carpentry and bridge and wine and knitting and brass-rubbing and line-dancing and the hundreds of other activities that enrich and enliven the daily toil of getting and spending, mortgages and shopping, school and office. There are rules, conventions, techniques, reserved objects, equipment and paraphernalia, time-honoured modes, forms, jargon and tradition. The average practitioner doesn’t expect to win prizes, earn a fortune, become famous or acquire absolute mastery in their art, craft, sport-or as we would say now, their chosen leisure pursuit. It really is enough to have fun.

The point remains: it isn’t a burden to learn the difference between acid and alkaline soil or understand how f-stops and exposure times affect your photograph. There’s no drudgery or humiliation in discovering how to knit, purl and cast off, snowplough your skis, deglaze a pan, carve a dovetail or tot up your bridge hand according to Acol. Only an embarrassed adolescent or deranged coward thinks jargon and reserved languages are pretentious and that detail and structure are boring. Sensible people are above simpering at references to colour in music, structure in wine or rhythm in architecture. When you learn to sail you are literally shown the ropes and taught that they are called sheets or painters and that knots are hitches and forward is aft and right is starboard. That is not pseudery or exclusivity, it is precision, it is part of initiating the newcomer into the guild. Learning the lingo is the beginning of our rite of passage.

In music, tempo is not the same as rhythm, which is not the same as pulse. There are metronomic indications and time signatures. At some point along the road between picking out a tune with one finger and really playing we need to know these distinctions. For some it comes naturally and seems inborn, for most of us the music is buried deep inside but needs a little coaxing and tuition to be got out. So someone shows us, or we progress by video, evening class or book. Talent is inborn but technique is learned.

Talent without technique is like an engine without a steering wheel, gears or brakes. It doesn’t matter how thoroughbred and powerful the V12 under the bonnet if it can’t be steered and kept under control. Talented people who do nothing with their gifts often crash and burn. A great truth, so obvious that it is almost a secret, is that most people are embarrassed to the point of shame by their talents. Ashamed of their gifts but proud to bursting of their achievements. Do athletes boast of their hand-eye coordination, grace and natural sense of balance? No, they talk of how hard they trained, the sacrifices they made, the effort they put in.

Ah, but a man’s reach should exceed his grasp
Or what’s a heaven for?

Robert Browning’s cry brings us back, at last, to poetry. While it is perfectly possible that you did not learn music at school, or drawing and painting, it is almost certain that you did learn poetry. Not how to do it, almost never how to write your own, but how, God help us, to appreciate it.

We have all of us, all of us, sat with brows furrowed feeling incredibly dense and dumb as the teacher asks us to respond to an image or line of verse.

What do you think Wordsworth was referring to here?
What does Wilfred Owen achieve by choosing this metaphor?
How does Keats respond to the nightingale?
Why do you think Shakespeare uses the word ‘gentle’ as a verb?
What is Larkin’s attitude to the hotel room?

It brings it all back, doesn’t it? All the red-faced, blood-pounding humiliation and embarrassment of being singled out for comment.

The way poetry was taught at school reminded W. H. Auden of a Punch cartoon composed, legend has it, by the poet A. E. Housman. Two English teachers are walking in the woods in springtime. The first, on hearing birdsong, is moved to quote William Wordsworth:

TEACHER 1: Oh cuckoo, shall I call thee bird
Or but a wandering voice?
TEACHER 2: State the alternative preferred

Even if some secret part of you might have been privately moved and engaged, you probably went through a stage of loathing those bores Shakespeare, Keats, Owen, Eliot, Larkin and all who came before and after them. You may love them now, you may still hate them or perhaps you feel entirely indifferent to the whole pack of them. But however well or badly we were taught English literature, how many of us have ever been shown how to write our own poems?

Don’t worry, it doesn’t have to rhyme. Don’t bother with metre and verses. Just express yourself. Pour out your feelings.

Don’t worry, just lift the lid and express yourself. Pour out your feelings.

We have all heard children do just that and we have all wanted to treat them with great violence as a result. Yet this is the only instruction we are ever likely to get in the art of writing poetry: Anything goes.

But that’s how modern poetry works, isn’t it? Free verse, don’t they call it? Vers libre?

Ye-e-es…And in avant-garde music, John Cage famously wrote a piece of silence called ‘4 Minutes 33 Seconds’ and created other works requiring ball-bearings and chains to be dropped on to prepared pianos. Do music teachers suggest that to children? Do we encourage them to ignore all harmony and rhythm and just make noise? It is important to realise that Cage’s first pieces were written in the Western compositional tradition, in movements with conventional Italian names like lento, vivace and fugato. Picasso’s early paintings are flawless models of figurative accuracy. Listening to music may inspire an extraordinary emotional response, but extraordinary emotions are not enough to make music.

Unlike musical notation, paint or clay, language is inside every one of us. For free. We are all proficient at it. We already have the palette, the paints and the instruments. We don’t have to go and buy any reserved materials. Poetry is made of the same stuff you are reading now, the same stuff you use to order pizza over the phone, the same stuff you yell at your parents and children, whisper in your lover’s ear and shove into an e-mail, text or birthday card. It is common to us all. Is that why we resent being told that there is a technique to its highest expression, poetry? I cannot ski, so I would like to be shown how to. I cannot paint, so I would value some lessons. But I can speak and write, so do not waste my time telling me that I need lessons in poetry, which is, after all, no more than emotional writing, with or without the odd rhyme. Isn’t it?

Jan Schreiber in a review of Timothy Steele’s Missing Measures, says this of modern verse:

The writing of poetry has been made laughably easy. There are no technical constraints. Knowledge of the tradition is not necessary, nor is a desire to communicate, this having been supplanted in many practitioners by the more urgent desire to express themselves. Even sophistication in the manipulation of syntax is not sought. Poetry, it seems, need no longer be at least as well written as prose.

Personally, I find writing without form, metre or rhyme not ‘laughably easy’ but fantastically difficult. If you can do it, good luck to you and farewell, this book is not for you: but a word of warning from W.H. Auden before you go.

The poet who writes ‘free’ verse is like Robinson Crusoe on his desert island: he must do all his cooking, laundry and darning for himself. In a few exceptional cases, this manly independence produces something original and impressive, but more often the result is squalor—dirty sheets on the unmade bed and empty bottles on the unswept floor.

I cannot teach you how to be a great poet or even a good one. Dammit, I can’t teach myself that. But I can show you how to have fun with the modes and forms of poetry as they have developed over the years. By the time you have read this book you will be able to write a Petrarchan sonnet, a Sapphic Ode, a ballade, a villanelle and a Spenserian stanza, among many other weird and delightful forms; you will be confident with metre, rhyme and much else besides. Whether you choose to write on the stupidity of advertising, the curve of your true love’s buttocks, the folly of war or the irritation of not being able to open a pickle jar is unimportant. I will give you the tools, you can finish the job. And once you have got the hang of the forms, you can devise your own. The Robertsonian Sonnet. The Jonesian Ode. The Millerian Stanza.

This is not an academic book. It is unlikely to become part of the core curriculum. It may help you with your English exams because it will certainly allow you to be a smart-arse in Practical Criticism papers (if such things still exist) and demonstrate that you know a trochee from a dactyl, a terza from an ottava rima and assonance from enjambment, in which case I am happy to be of service. It is over a quarter of a century since I did any teaching and I have no idea if such knowledge is considered good or useless these days, for all I know it will count against you.

I have written this book because over the past thirty-five years I have derived enormous private pleasure from writing poetry and like anyone with a passion I am keen to share it. You will be relieved to hear that I will not be burdening you with any of my actual poems (except sample verse specifically designed to help clarify form and metre): I do not write poetry for publication, I write it for the same reason that, according to Wilde, one should write a diary, to have something sensational to read on the train. And as a way of speaking to myself. But most importantly of all for pleasure.

This is not the only work on prosody (the art of versification) ever published in English, but it is the one that I should like to have been available to me many years ago. It is technical, yes, inasmuch as it investigates technique, but I hope that does not make it dry, obscure or difficult-after all, ‘technique’ is just the Greek for ‘art’. I have tried to make everything approachable without being loopily matey or absurdly simplistic.

I certainly do not attempt in this book to pick up where those poor teachers left off and instruct you in poetry appreciation. I suspect, however, that once you have started writing a poem of any real shape you will find yourself admiring and appreciating other poets’ work a great deal more. If you have never picked up a golf club you will never really know just how remarkable Ernie Els is (substitute tennis racket for Roger Federer, frying pan for Gordon Ramsay, piano for Jools Holland and so on).

But maybe you are too old a dog to learn new tricks? Maybe you have missed the bus? That’s hooey. Thomas Hardy (a finer poet than he was a novelist in my view) did not start publishing verse till he was nearly sixty.

Every child is musical. Unfortunately this natural gift is squelched before it has time to develop. From all my life experience I remember being laughed at because my voice and the words I sang didn’t please someone. My second grade teacher, Miss Stone would not let me sing with the rest of the class because she judged my voice as not musical and she said I threw the class off key. I believed her which led to the blockage of my appreciation of music and blocked my ability to write poetry. Fortunately at the age of 57 I had a significant emotional event which unblocked my ability to compose poetry which many people believe has lyrical qualities.

So writes one Sidney Madwed. Mr Madwed may not be Thomas Campion or Cole Porter, but he believes that an understanding of prosody has set him free and now clearly has a whale of a time writing his lyrics and verses. I hope reading this book will take the place for you of a ‘significant emotional event’ and awaken the poet that has always lain dormant within.

It is never too late. We are all opsimaths.

Opsimath, noun: one who learns late in life.

Let us go forward together now, both opsimathically and optimistically. Nothing can hold us back. The ode beckons.

Written by S

Sun, 2013-08-04 at 18:27:35 +05:30

Posted in language, literature, quotes

## The functional equation f(x+y) = f(x)f(y)

Suppose $f: \mathbb{R} \to \mathbb{R}$ satisfies $f(x+y) = f(x) f(y)$. What can we say about $f$?

Putting $y = 0$ gives

$\displaystyle f(x) = f(x+0) = f(x)f(0),$

which can happen if either $f(x) = 0$ or $f(0) = 1$. Note that the function $f$ which is identically zero satisfies the functional equation. If $f$ is not this function, i.e., if $f(x) \neq 0$ for at least one value of $x$, then plugging that value of $x$ (say $x^*$) into the equation gives $f(0) = 1$. Also, for any $x$, the equation $f(x^*) = f(x +x^* - x) = f(x)f(x^* - x)$ forces $f(x) \neq 0$ as well. Further, $f(x) = f(x/2 + x/2) = f(x/2)^2$ so $f(x) > 0$ for all $x$.

Next, putting $y = x$ gives $f(2x) = f(x)^2$, and by induction $f(nx) = f(x)^n$. Putting $\frac{x}{n}$ in place of $x$ in this gives $f(n\frac{x}{n}) = f(\frac{x}{n})^n$ which means $f(\frac{x}{n}) = f(x)^{\frac1n}$ (note we’re using $f(x) > 0$ here). And again, $f(\frac{m}{n}x) = f(x)^{m/n}$. So $f(\frac{m}{n}) = f(1)^{m/n}$, which completely defines the function at rational points.

[As $f(1) > 0$, it can be written as $f(1) = e^k$ for some constant $k$, which gives $f(x) = e^{kx}$ for rational $x$.]

To extend this function to irrational numbers, we need some further assumptions on $f$, such as continuity. It turns out that being continuous at any point is enough (and implies the function is $f(x) = f(1)^x$ everywhere): note that $f(x + m/n) = f(x)f(m/n) = f(x)f(1)^{m/n}$. Even being Lebesgue-integrable/measurable will do.

Else, there are discontinuous functions satisfying the functional equation. (Basically, we can define the value of the function separately on each “independent” part. That is, define the equivalence class where $x$ and $y$ are related if $y = r_1x + r_2$ for rationals $r_1$ and $r_2$, pick a representative for each class using the axiom of choice (this is something like picking a basis for $\mathbb{R}/\mathbb{Q}$, which corresponds to the equivalence class defined by the relation $y = r_1x$), define the value of the function independently for each representative, and this fixes the value of $f$ on $\mathbb{R}$. See this article for more details.)

To step back a bit: what the functional equation says is that $f$ is a homorphism from $(\mathbb{R}, +)$, the additive group of real numbers, to $(\mathbb{R}, \times)$, the multiplicative monoid of real numbers. If $f$ is not the trivial identically-zero function, then (as we saw above) $f$ is in fact a homomorphism from $(\mathbb{R}, +)$, the additive group of real numbers, to $(\mathbb{R_+^*}, \times)$, the multiplicative group of positive real numbers. What we proved is that the exponential functions $e^{kx}$ are precisely all such functions that are nice (nice here meaning either measurable or continuous at least one point). (Note that this set includes the trivial homomorphism corresponding to $k = 0$: the function $f(x) = 1$ identically everywhere. If $f$ is not this trivial map, then it is in fact an isomorphism.)

Written by S

Mon, 2013-04-08 at 11:24:08 +05:30

Posted in mathematics

## Trajectory of a point moving with acceleration perpendicular to velocity

(Just some basic high-school physics stuff; to assure myself I can still do some elementary things. :P Essentially, showing that if a particle moves with acceleration perpendicular to velocity, or velocity perpendicular to position, then it traces out a circle. Stop reading here if this is obvious.)

Suppose a point moves in the plane such that its acceleration is always perpendicular to its velocity, and of the same magnitude. What is its path like?

To set up notation: let’s say the point’s position at time $t$ is $(p_x(t), p_y(t))$, its velocity is $(v_x(t), v_y(t)) = \left(\frac{d}{dt}p_x(t), \frac{d}{dt}p_y(t)\right)$, and its acceleration is $(a_x(t), a_y(t)) = \left(\frac{d}{dt}v_x(t), \frac{d}{dt}v_y(t)\right)$.

The result of rotating a point $(x,y)$ by 90° is $(-y, x)$. (E.g. see figure below)

So the fact that acceleration is at right angles to velocity means that $(a_x(t), a_y(t)) = (-v_y(t), v_x(t))$, or, to write everything in terms of the velocity,

\begin{aligned} \frac{d}{dt}v_x(t) &= -v_y(t) \\ \frac{d}{dt}v_y(t) &= v_x(t) \end{aligned}

where we can get rid of $v_x(t)$ by substituting the second equation (in the form $v_x(t) = \frac{d}{dt}v_y(t)$) into the first:

$v_y(t) = -\frac{d}{dt}v_x(t) = -\frac{d}{dt}\left(\frac{d}{dt}v_y(t)\right)$

or in other words

$v_y(t) = -\frac{d^2}{dt^2}v_y(t).$

By some theory about ordinary differential equations, which I don’t know (please help!) (but see the very related example you saw in high school, of simple harmonic motion), the solutions to this equation are $\sin(t)$ and $\cos(t)$ and any linear combination of those: the solution in general is

\begin{aligned} v_y(t) &= a \sin(t) + b \cos(t) \\ &= \sqrt{a^2 + b^2} \left(\frac{a}{\sqrt{a^2+b^2}}\sin(t) + \frac{b}{\sqrt{a^2+b^2}}\cos(t)\right) \\ &= R\sin (t + \alpha) \end{aligned}

where $R = \sqrt{a^2 + b^2}$ and $\alpha$ is the angle such that $\cos(\alpha) = \frac{a}{\sqrt{a^2+b^2}}$ and $\sin(\alpha) = \frac{b}{\sqrt{a^2+b^2}}$. And the fact that $v_x(t) = \frac{d}{dt}v_y(t)$ gives $v_x(t) = R\cos(t + \alpha)$. So $(v_x(t), v_y(t)) = (R\cos(t + \alpha), R\sin(t + \alpha))$. Note that $(a_x(t), a_y(t)) = \left(\frac{d}{dt}v_x(t), \frac{d}{dt}v_y(t)\right) = (-R\sin(t+\alpha), R\cos(t+\alpha))$ is indeed perpendicular to $(v_x(t), v_y(t))$ as we wanted.

The actual trajectory $(p_x(t), p_y(t))$ can be got by integrating

$\left(\frac{d}{dt}p_x(t), \frac{d}{dt}p_y(t)\right) = (v_x(t), v_y(t)) = (R\cos(t + \alpha), R\sin(t + \alpha))$

to get $p_x(t) = R\sin(t + \alpha) + c_1$ and $p_y(t) = -R\cos(t + \alpha) + c_2$. This trajectory is a point moving on a circle centered at point $(c_1, c_2)$ and of radius $R$, with speed $R$ or unit angular speed. Note that velocity is also perpendicular to the point’s position wrt the centre of the circle, as velocity is tangential to the circle, as it should be.

With a suitable change of coordinates (translate the origin to $(c_1, c_2)$, then rotate the axes by $\frac{\pi}{2}+\alpha$, then scale everything so that $R = 1$), this is the familiar paremetrization $(\cos(t), \sin(t))$ of the circle.

Note: Just as we derived $(v_x(t), v_y(t)) = (R\cos(t + \alpha), R\sin(t + \alpha))$ from assuming that the acceleration is perpendicular to velocity, we can, by assuming that velocity is perpendicular to position, identically derive $(p_x(t), p_y(t)) = (R\cos(t + \alpha), R\sin(t + \alpha))$, i.e. that the point moves on a circle.

Written by S

Sun, 2013-04-07 at 23:38:01 +05:30

Posted in mathematics

## Typing Kannada on Mac OS X

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(Thanks to this and this.)

Turns out it’s very easy, and we can basically use the same input method (UIM) as in Linux.

1. Get MacUIM from its website
2. Install it.
3. Go to System Preferences -> Language & Text -> Input Sources, and turn on MacUIM. Tick “Show Input menu in menu bar” too.
4. I now have three input methods: US, EasyIAST (see earlier post), and MacUIM (Roman).
5. Go to System Preferences -> MacUIM -> General, and in Input method, choose m17n-kn-itrans
6. Go to System Preferences -> MacUIM -> Helper, tick “Use Helper-Applet”, and in the list at the right, tick m17n-kn-itrans.
7. [Just for me] I have some changes to kn-itrans.mim, to make it closer to HK (and remove nonsense like “RRi” or whatnot just to type ಋ): download this file kn-itrans.mim, and remove the pdf extension. It goes into /Library/M17NLib/share/m17n/kn-itrans.mim

Written by S

Sun, 2013-04-07 at 01:14:35 +05:30

Posted in compknow

with one comment

These used to be ubiquitous a while ago (IIRC, I used to carry one of these daily to school as my lunch basket at some point; we still have one such basket at home), but photos seem hard to find on the internet (or I’m just missing the right keywords). So, photos:

Poorani Ammal, Copyright R Revathi / Mylapore Times

Written by S

Sun, 2013-04-07 at 00:14:29 +05:30

Posted in Uncategorized

## The power series for sin and cos

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There are many ways to derive the power series of $\sin x$ and $\cos x$ using the machinery of Taylor series etc., but below is another elementary way of demonstrating that the well-known power series expansions are the right ones. The argument below is from Tristan Needham’s Visual Complex Analysis, which I’m reproducing without looking at the book just to convince myself that I’ve internalized it correctly.

So: let
\displaystyle \begin{aligned} P(x) &= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots \quad \text{ and }\\ Q(x) &= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots . \end{aligned}

We will take the following two for granted (both can be proved with some effort):

1. Both power series are convergent.
2. The power series can be differentiated term-wise.

As suggested by (2) above, the first thing we observe is that $\frac{d}{dx}P(x) = Q(x)$ and $\frac{d}{dx}Q(x) = -P(x)$.

So firstly:
\begin{aligned} \frac{d}{dx}(P(x)^2 + Q(x)^2) &= 2P(x)P'(x) + 2Q(x)Q'(x) \\ &= 2P(x)Q(x) - 2Q(x)P(x) \\ &= 0 \end{aligned}
which means that $P(x)^2 + Q(x)^2$ is a constant and does not vary with $x$. Putting $x = 0$ shows that $P(0) = 0$ and $Q(0) = 1$, so $P(x)^2 + Q(x)^2 = 1$ for all $x$.

Secondly, define the angle $\theta$ as a function of $x$, by $\tan \theta(x) = P(x)/Q(x)$. (To be precise, this defines $\theta(x)$ up to a multiple of $\pi$, i.e. modulo $\pi$.)
Differentiating the left-hand side of this definition gives
\displaystyle \begin{aligned} \frac{d}{dx} \tan \theta(x) &= (1 + \tan^2 \theta(x)) \theta'(x) \\ &= (1 + \frac{P(x)^2}{Q(x)^2}) \theta'(x) \\ &= \frac{1}{Q(x)^2} \theta'(x) \end{aligned}
(where $\theta'(x)$ means $\frac{d}{dx} \theta(x)$)
while differentiating the right-hand side gives
\displaystyle \begin{aligned} \frac{d}{dx} \frac{P(x)}{Q(x)} &= \frac{Q(x)P'(x) - P(x)Q'(x)}{Q(x)^2} \\ &= \frac{Q(x)Q(x) + P(x)P(x)}{Q(x)^2} \\ &= \frac{1}{Q(x)^2} \end{aligned}

The necessary equality of the two tells us that $\frac{d}{dx}\theta(x) = 1$, which along with the initial condition $\tan \theta(0) = P(0)/Q(0) = 0 = \tan 0$ that says $\theta(0) \equiv 0 \mod \pi$, gives $\theta(x) = x$ (or to be precise, $\theta(x) \equiv x \pmod {\pi}$).

In other words, we have shown that the power series $P(x)$ and $Q(x)$ satisfy $\frac{P(x)}{Q(x)} = \tan x = \frac{\sin x}{\cos x}$ and therefore $P(x) = k \sin x$ and $Q(x) = k \cos x$ for some $k$. The observation that $Q(0) = 1 = \cos 0$ (or our earlier observation that $P(x)^2 + Q(x)^2 = 1$ for all $x$) gives $k = 1$, thereby showing that $P(x) = \sin x$ and $Q(x) = \cos x$.

So much for $\sin x$ and $\cos x$. Just as an aside, observe that if we take $i$ to be a symbol satisfying $i^2 = -1$, then
\displaystyle \begin{aligned} \cos x + i\sin x &= Q(x) + iP(x) \\ &= \left(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots\right) + i\left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots \right) \\ &= 1 + ix + \frac{-x^2}{2!} + \frac{-ix^3}{3!} + \frac{x^4}{4!} + \frac{ix^5}{5!} + \frac{-x^6}{6!} + \frac{-ix^7}{7!} + \dots \\ &= 1 + ix + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \frac{(ix)^4}{4!} + \frac{(ix)^5}{5!} + \frac{(ix)^6}{6!} + \frac{(ix)^7}{7!} + \dots \end{aligned}
the right hand side of which looks very much like the result of “substituting” $y = ix$ in the known (real) power series
$\displaystyle e^y = 1 + y + \frac{y^2}{2!} + \frac{y^3}{3!} + \dots$
(which itself can be proved using the term-wise differentiation above and the defining property $\frac{d}{dx} e^x = e^x$, say).

So this is one heuristic justification for us to define $e^{ix} = \cos x + i\sin x$.
Or, if we define $e^{ix}$ as the result of substituting $ix$ in the real power series for $e^y$, this proves that $e^{ix} = \cos x + i\sin x$.

Written by S

Fri, 2013-03-08 at 00:33:12 +05:30

Posted in mathematics

## A better keyboard layout for typing IAST on Mac OS X (based on EasyUnicode)

To type IAST (English letters with diacritics, for Sanskrit transliteration) on Mac OS X, perhaps the easiest way, rather than to use transliteration tools, is to get a keyboard layout that does it. Just to be clear, this is the alphabet we want:

a ā i ī u ū ṛ ṝ ḷ ḹ e ai o au ṃ ḥ
k kh g gh ṅ
c ch j j ñ
ṭ ṭh ḍ ḍh ṇ
t t d dh n
p ph b bh m
y r l v ś ṣ s h


In other words, the special characters needed are:

• Letters with macron above: ā ī ū ṝ ḹ plus it may be occasionally useful to have ē and ō as well
• Letters with dot below: ṭ ḍ ṇ ṣ (the retroflex consonants), also the vowels ṛ ṝ ḷ ḹ, plus ṃ and ḥ (anusvāra)
• Letters with other marks above: ṅ ñ ś

There is a keyboard layout that does this: It’s called “EasyUnicode”, created by Toshiya Unebe (Nagoya University), and is documented at http://ebmp.org/p_easyunicode.php (“EasyUnicode version 5″ it says) (PDF version), and you can download it from http://www.ebmp.org/p_dwnlds.php (EBMP) (=Early Buddhist Manuscripts Project, University of Washington) or also http://www.palitext.com/subpages/PC_Unicode.htm “Pali Fonts for PC and Unicode”. (Page in Japanese.)

This keyboard layout is just like the usual (US English) layout ordinarily, but when you hold down the Alt (Option) key and press a, you get ā, similarly Option+s gives ś, Option+n gives ñ and Option+g gives ṅ, etc. The full mapping is available along with other documentation in the download above.

This is very convenient. One issue with the layout is that also overrides a lot of keys for no apparent reason (Ctrl-A / Ctrl-E etc. stopped working for me), so I got Ukelele from SIL, and wrote my own keyboard layout. I’ve called it EasyIAST, and it is available here for now. I plan to add a README etc. and distribute it in some proper way later; for now you can use the instructions from EasyUnicode above. If you find it useful and/or make any improvements, please let me know as well.

If some time is available, it would be good to make a Devanagari keyboard layout along the same lines.

Written by S

Tue, 2013-01-22 at 11:28:55 +05:30

Posted in sanskrit

Tagged with , ,