Mathematical Insights--Infinite

sums,exponential function,Catenary and

Leonhard Euler

Dr N K Srinivasan

Introduction

Leonhard Euler (1707-1783) was one of those all time greats in

mathematics. While he traversed many fields in

mathematics--that is , algebra,geometry and analysis

[calculus], his special efforts seemed to be in the area of

infinite sums and infinite series. He was indeed a prolific

mathematician in a long career. His equation ,known as "Euler

Equation " is one of the corner stones of modern mathematics.

Euler and some infinite series

Euler was attracted to trigonometric functions such as sine and

cosine of angles.

You know that if you know sine tables ,that is enough.You can

generate cosine of angles because:

sin2 (Θ) + cos2 (Θ) = 1

Euler made use of the infinite series expansion of sine

function in many works:

sin x = 1 - x3 /3! + x5 /5! - x7 /7! ------

Euler made use of these expansions ingeniously.!

What is ' e1 ' ?

You must have come across 'e' ,the base of the natural

logarithm earlier.

If y = ln (x) , then ey = x

The quantity 'e' is again the sum of infinite terms:

e = 1 + 1/1! + 1/ 2! + 1/3! + 1/ 4! + 1/5!

....-------

It is easy to calculate the first few terms and add them up:

e = 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120

= 1 + 1 + 0.5 + 0. 1666 + 0.0416 + 0.0083

= 2.7165 up to the sixth term

Well, this series converges fast ! What we mean by that? Each

term adds less and less quantity to the previous sum and soon

the sum tends to the limit : e = 2.718281828459....

Exponential Function

The function y = ex = exp(x) is justifiably the most famous

function you will be knowing. The properties of this function

can fill a separate book.! Let us explore a few of its

interesting features , besides hundreds of applications that

exist , such as population studies and radio-active decay.!

We can expand exp(x) into an infinite series:

ex = 1 + x/1! + x2 /2! + x3 /3! +.......

-------equation(1)

If we put x = 1, we get the series for 'e', as given above!

---simple indeed.This series converges.

Note that if x <= 1, the series would converge , but not

otherwise.!

A more useful form of this function is as follows:

y = ekx

where k is a constant.

You can explore different cases of this function:

1. Take k as positive and >1 ; k= 2 --> y = exp(2x) Plot

the graph;it rises very rapidly.

This could model the population growth of some countries and x

is ,of course, t= years.

2 Take k as negative and < 1: k= -0.01 ---> y = exp(-0.01x)

Plot the graph; it decays rapidly.

Again take x = t= time in years. You can model the radioactive

decay of a radioactive isotopes with suitable decay constant or

half life.

Euler Formula

Euler went some steps ahead and put x = iΠ .

You know that pi is a transcendental number and is just the

ratio of circumference of a circle to its diameter.

What we would get if we put iΠ as the exponent in this

equation?

Euler had a surprise:

exp(iπ) = cos(π) + i sin (π) ----------(equation 2)

This is one of the beautiful relations in Mathematics----

according to Richard Feynman.

Why? it links up the exponential with the trig functions ; it

contains the imaginary number i; you know that

i.i = -1.

There are many uses of this relation.

This relation extends to complex functions too.Each term can be

treated as a vector and added.

How to prove equation 2?

Euler gave a very simple proof. Expand both sides of the

equation by infinite series and see for yourself:

By equation 1, we get

exp(iπ) = 1 + (iπ) + (iπ)^2/2 + (iπ)^3/6 + (iπ)^4/24 +

-----

Note that i^2 = -1, i^3= -i i^4 = +1 and so on.

exp( iπ) = 1 + (i π) - (π)^2/2 -i(π)^3/6 + (π)^4/24

+-----

= [ 1- π^2/2 + π^4/24 ----] + i [ π - π^3/6 +

π^5/120 ---]

= cos (π) + i sin (π)

Now the first term is -1 and the second term = 0

So we get : exp(iπ) + 1 = 0 equation (3)

Equation (3) is the epoch-making equation in mathematics. You

will find many applications as you learn advanced math.

Exponential functions have other great applications. Let us

turn to one of the fascinating applications.

Hanging Chain, Bernoulli Brothers and the Gateway Arch

of St.Louis, Missouri

What is the function that describes the shape of a hanging

chain or cable between two supports or towers.

This is not a parabola, but pretty close to it.

Galileo examined this and wrote that though it is not a

parabola, it approximates to a parabola.

What exactly is the equation for a hanging chain?

We shall call this curve "Catenary". Historians say that this

name was given by Thomas Jefferson.!Some say by Huygens.

This problem was posed by Jakob Bernoulli(1654-1705), the elder

brother of the two Bernoulli brothers, both eminent math

wizards . The younger one, Johann (1667-1748) solved the

problem overnight. Apparently the elder Bernoulli was

struggling for nearly a year.![Leibniz and Christiaan Huygens

also solved it].

Jacob Bernoulli

The two brothers always had rivalry in mathematical work and

challenged each other and showed one -up manship. They were not

so cooperative as the two brothers --Orville and Wilbur Wright

of aviation fame. After solving this catenary problem, Johann

wrote ; " the efforts of my brother were without success....but

the next morning ,filled with joy, I ran to my brother,who was

still struggling miserably with this Gordian knot without

getting anywhere...."

The curve is a "cosine hyperbolic" function:

y = a Cosh (x/a)

This can be written as a sum of increasing

exponentialfor positive x and decreasing exponential

functions:

------------------equation (4)

Note that this equation has only one parameter, 'a'. By

changing this 'a', we can get different catenaries and choose

one that is aesthetically satisfying or structurally easy to

build.

Practical Application of Catenary

This curve is used for building cathedral arches, the

gateway arch ,that is "gateway to the west",in St Louis,

Missouri ,and so on.

Robert Hooke used it in rebuilding the arches of St Paul's

Cathedral in London.He wrote in a cryptic manner: "As hangs the

chain, so stands the arch." The arch is ,of course, an inverted

hanging chain or catenary.

David Gregory wrote a treatise on this in 1697.

Euler showed that the curve rotated about the X axis gives a

surface of minimum area called "Catenoid".

Gateway Arch at St Louis

What about the Gateway Arch in St Louis , Missouri, designed by

the great architect, Euro Saarinen ?

It was thought that this one is a catenary. Now it is found to

be a modified or flattened Catenary, following an equation of

the type:

y = A cosh (Bx) + C

where B is not 1/A. In this arch, A = 0.69/B. This is also

called a weighted catenary.

Another view of Gateway Arch

A recent paper by Robert Osserman, an emeritus professor of

Math from Stanford Univ, explores this and offers insight into

the use of Catenary. [ "Notices of Amer math Society , Feb,

2010"].

It is obvious that an architect may modify a mathematical

expression to design an aesthetically pleasing arch.Further a

physical arch is a three-dimensional body with the thickness

section being designed for structural integrity. The catenary

equation is a curve.The Arch has finite thickness with a

cross--sectional shape. The physical arch should be

structurally stable , strong and easy to build.

Euro Saarinen went on to design several structures with

catenary curve,suitably modified in some way.

You have seen many electrical cables with the hanging chain

configuration. Much use of the catenary equation is made for

designs of these structures.

Euler and Bernoulli Connection

Leonhard Euler was born and grew up in Basel,Switzerland. His

father taught him elementary math at home.He had mostly private

tutors. At the age of 14, he entered the university at Basil.

Math was not taught there. As it was common those days , Euler

studied theology, languages and medicine. His love for math

blossomed because he took saturday afternoon tutoring with the

great mathematician ,Johann Bernoulii.

He also became friends with the two sons of Johann-- Nicolas

and Daniel Bernoullis , who became famous later.

After completing the university, Euler could not get any

teaching position in math. Fortunately, Nicolas and Daniel had

moved to Russia and were well supported at the Russian Academy

of ciences at St Petersburg, often called by mathematicians as

simply Petersburg Academy.It was started by Peter the Great of

Russia and his second wife,Catherine I , both enlightened

persons. Peter invited many bright men to the Royal court and

the Academy. At the instance of Nicolas and Daniel Bernoulli,

Euler was invited.

Euler arrived in St Petersburg in 1927 ---only to find that the

Czar and his queen were no more. He spent the next 14 years in

the company of great mathematicians and physicists . His output

was so prolific that the Academy Journal could publish only a

'random sampling' of his works.

Euler got the invitation from the Berlin Academy, founded by

another great monarch, Frederick, the great .Euler left Russia

in 1741 for Berlin. Though his output in math was unabated, he

was not liking the royal court with many sycophants and snobs.

Euler was a taciturn man with very few words for the monarch.

As luck could have it, Euler again had an invitation from

Russia, this time by Catherine II, Catherine the Great . This

was in 1766, 15 years after he went to Berlin. Euler continued

his work at St Petersburg .

His eyesight failed and he wrote on a black-board his

equations. His children and grand-children copied them down in

notebooks.

There was a fire in his palatial house in 1771. Fortunately a

kindly assistant lifted him out of the house.Euler continued

his math work ,till the very end. He died in 1783.

A major work of Euler is his book on Analysis or calculus ,

entitled " An Introduction to Infinite Analysis" . This is a

milestone in the annals of mathematics ,like Euclid's 'Elements

' and Newton's

"Principia".

Euler's work extends to other areas too. His work in topology

is highly significant.He solved the Konisberg bridge problem.

Some infinite series---teacher and student

Euler's work included several infinite series of great

interest.

The Harmonic series, H ,which is the sum of reciprocals of

integers ,does not converge:

H = 1 + 1/2 + 1/3 + 1/4 -----

This was proved by Jakob Bernoulli, the elder brother of the

two.

But the younger one , Johann posed a challenge problem:

Find the sum: S= 1+ 1/4 + 1/9 + 1/16...... to infinity.

He knew pretty well that his brother would not be able to solve

this.! So, he gave a challenge to whosoever interested with the

statement:

" If anyone finds and communicates to us that which upto now

has eluded our efforts, great will be our gratitude."

Jakob has previously shown that the series converges and the

sum is less than 2.

Euler took up the problem and solved it :

S = 1 + 1/4 + 1/9 + 1/16 +........=π2 /6

Since pi = 3.1416, S = 1.6449

A proof is given in ref 6 by William Dunham.]

We can use the above relation to calculate pi;[May be, you can

write a small computer program for this and run it!]. It

converges slowly ,though.The first ten terms add up to 1.5497.

Johann Bernoulli must have been delighted to find this proof

from his former student Euler whom he coached in saturday

afternoons.!

Euler Formula and the complex number

A complex number is written as follows: z = a + i b

Then the magnitude of this number : |z | = r =√ (a² + b²)

Now a= rcosθ and b= r sinθ

Using Euler formula: z= r exp (iθ) = r (cos θ + i sin θ)

A great advantage is that multiplication and division become

easy to do:

z1 .z2 = r1 .r2 exp(i (θ1 + θ2))

You can write the expression for z1 / z2 ,using Euler formula.

References

1 E T Bell Men of Mathematics--- Dover Pub, NY

2 James R Newman The world of mathematics --Vol 1 .... Simon

& Schuster, N Y

3 Ruel V Churchill Functions of a complex variable-- McGraw

Hill

4 Robert Osserman -- Notices of American Math Society, Feb

2010.

5 Robert P Crease -- The great equations--W W Norton & co, NY

6 William Dunham -- Journey through genius--Penguin, NY

[Note: The author can be contacted at :

nksrinivasan@hotmail.com ]

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