euler's marvelous formula

7/25/2019 Euler's Marvelous Formula

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Elementary Functions

Part 5, Advanced TrigonometryLecture 5.7a, Eulers Marvelous Formula

Dr. Ken W. Smith

Sam Houston State University

2013

Smith (SHSU) Elementary Functions 2013 1 / 14

Eulers Equation

The value of complex numbers was recognized but poorly understoodduring the late Renaissance period (1500-1700 AD.) The number systemwas explicitly studied in the late 18th century. Eulerusedi for the square

root of1 in 1779. Gaussused the term complex in the early 1800s.The complex plane (Argand diagram or Gauss plane) was introducedin a memoir by Argand in Paris in 1806, although it was implicit in thedoctoral dissertation of Gauss in 1799 and in work of Caspar Wesselaround the same time.


Eulers Equation

Notice the following remarkable fact that if

z =

3

2 +

1

2i= cos

6+i sin

6

thenz3

=i. (Multiply it out & see!) Thus z12

= 1 and so z is a twelfthroot of 1.Now the polar coordinate form for z is r = 1, =

6, that is, z is exactly

one-twelfth of the way around the unit circle. z is a twelfth root of 1 and itis one-twelfth of the way around the unit circle. This is not a coincidence!DeMoivre apparently noticed this and proved (by induction, using sum ofangles formulas) that ifn is an integer then

(cos +i sin )n = cos n+i sin n. (1)

Thus exponentiation, that is raising a complex number to some power, isequivalent to multiplication of the arguments. Somehow the angles in thecomplex number act like exponents.


Eulers Equation

Euler would explain why that was true. Using the derivative and infiniteseries, he would show that

ei = cos +i sin (2)

By simple laws of exponents, (eiz)n =einz and so Eulers equationexplains DeMoivre formula.

This explains the coincidence we noticed with the complex numberz= cos

6+i sin

6which is one-twelfth of the way around the unit circle;

raisingz to the twelfth power will simply multiply the angle by twelveand move the point z to the point with angle 2: (1, 0) = 1 + 0i.



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Trig functions in terms of the exponential function

Eulers formulaei = cos +i sin

allows us to write the exponential function in terms of the two basic trigfunctions, sine and cosine. We may then use Eulers formula to find aformula forcos z andsin z as a sum of exponential functions.By Eulers formula, with inputz,

eiz = cos(z) +i sin(z) = cos(z) i sin(z).Addthe expressions for eiz andeiz to get

eiz +eiz = 2 cos(z)

and so

cos z=eiz +eiz

2 . (3)

If we subtract the equationeiz = cos z i sin z from Eulers equationand then divide by 2i, we have a formula for sine:

sin z=eiz eiz

2i . (4)Smith (SHSU)

Elementary Functions 2013 5 / 14

Trig functions in terms of the exponential function

We wrote the exponential function in terms of cosine and sine

ei = cos +i sin

and then wrote the trig functions in terms of the exponential function!

cos z = eiz

+e

iz

2

sin z =eiz eiz

2i

The exponential and trig functions are very closely related. Trig functionsare, in some sense, really exponential functions in disguise!

And conversely, the exponential functions are trig functions!Smith (SHSU) Elementary Functions 2013 6 / 14

Some worked examples.

Lets try out some applications of Eulers formula. Here are some workedproblems.

Put the complex number z = ei in the Cartesian form z = a+bi.

Solution. z = ei = 1(cos() +i sin()) = 1(1 + 0i) = 1

It seems remarkable that if we combine the three strangestmathconstants,e, i and we get

ei = 1.

Some rewrite this in the form

e

i

+ 1 = 0

(often seen on t-shirts for engineering clubs or math clubs.)



Put the complex number z = 2e13

6 i in the Cartesian formz = a+bi.

Solution.z= 2e

13

6 i = 2 cos(13

6 ) + 2i sin(13

6 ) = 2cos(

6) + 2i sin(

6) =

3 +i.



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Put the complex number z = 18 + 26i in the polar form z = rei wherer, R and bothr and are positive.

Solution. The modulus ofz = 18 + 26i is 182 + 262 = 1000.So the polar coordinate form ofz = 18 + 26i is

103 ei where

= arctan(2618

). (The angle is about 0.96525166319.)



Find a cube root of the number z = 18 + 26iand put this cube root in theCartesian formz = a+bi. (Use a calculator and get an exact value forthis cube root.

Using the previous problem, we writez = 18 + 26i=

103 ei where= arctan(26

18).

The cube root of

103 ei is

10 ei

3

(The angle 3

is about 0.3217505544.) Using a calculator, we can see thatthis comes out to approximately

10 cos(

3) +i

10 sin(

3) = 3 +i.

One could check by computing (3 +i)3 and see that we indeed get

18 + 26i.Smith (SHSU) Elementary Functions 2013 10 / 14


Find a complex number z such that ln(1) = z.

Solutions. Since1 in polar coordinate form is1 = ei thenz = i is asolution toln(1).



A question found on the internet: What isii?

We can find one answer if we write the base i in polar form i = e

2i.

(More carefully, we might note that i = e

2i+2ki, for any integer k.)

Then ii = (e

2i)i =e

2i2 = e

2

0.207879576350761908546955619834978770033877841631769608075135...



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Complex numbers v. Real numbers

Here are some things one can do with the real numbers:

1 Show that f(x) = sinx is periodic with period2, that is,f(x+ 2) = f(x).

2 Find an infinite set of numbers, x, such that sin(x) = 1/2.

3 Find a number x such that ex = 200.

4 Compute ln(2).

Here are some things that require complex numbers:

1 Show that f(x) = ex is periodic with period 2i, that is,f(x+ 2i) =f(x).

2 Find an infinite set of numbers, x, such that ex = 1/2.

3 Find a number x such that sin(x) = 200.

4 Compute ln(2).

These are all topics for further exploration in a course in complex variables.Smith (SHSU) Elementary Functions 2013 13 / 14

Last Slide!

It is appropriate that we end our series of precalculus lectures with apresentation of Eulers marvelous formula, which brings together b oth thetrigonometric functions and the exponential functions into one form!

The applications of this formula appear in all the technology around us,and simplify many complicated mathematical computations!

ei = cos +i sin

(End)


euler's marvelous formula

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