Euler Identity values.

Question

e^iπ=-1,,,,where i=√(-1)

is called Euller's identity

This is the principle value of e^iπ.

What is the expression providing

the other values?

Answer 1

the answer is e^iπ=-e^2π*π*k,,,,,,k=0,+/-1,...

k=0, e^iπ=-1, k=1 e^iπ=-e^2π*π

let's look at q=z^w

z=|z|e^iθ=x+iy,,,|z|=(x²+y²)^(1/2),,,θ=tan^-1(y/x)

w=u+iv

z^w=e^w*lnz

lnz=ln|z|+i(θ+2πk),,,,,,,,,k=0,+/-1,...

from lnz=ln(|z|*e^i(θ+2πk))

z^w=e^{(u+iv)*(ln|z|+i(θ+2πk)}

z^w=[e^{(u*ln|z|-v(θ+2πk))}]*[e^{i(v*ln|z|+u(θ+2πk)+2πn)}],,,,n=0,+/-1,...

final answer for z^w

for q=e^iπ

z=e+0*i,,,,,|z|=e,,,θ=0

w=0+iπ,,,,,u=0, v=π

plug in and eliminate debris, change -k to k since k=0,+/-1,...

e^iπ=-e^(2π*π*k),,,,k=0,+/-1,...

Answer 2

e^iπ = e^{i(π + 2kπ)}, since sine and cosine are cyclic with a period of 2π.

= e^{iπ + 2kiπ}

= (e^iπ)(e^iπ)^2k

= (-1)(-1)^2k

= (-1)(1)^k

= -1, for all integral values of k.

Q.E.D.

Answer 3

e^ix = cosx + isinx

in general formula

if x=pi, then

e^ipi = cospi + isinpi = -1 +0= -1

when Euler first proved this, publicly, legend has it, it was at Catherine the Great's court, where invited Euler and an atheist philosopher Diderot to debate. After proving the idenity, Euler wrote QED, therefore God Exists. Diderot had no reply and left the court in disgrace.

It's sort of a version of Aquinas' Argument from Design. Mathematical Design implies a Designer. An imaginary power has real existence.

Agnostic Bertrand Russell called the Identity the most beautiful equation in mathematics. the beauty of numbers, as Ian Fleming in James Bond movies might phrase it. the equation relates 5 fundamental concepts in math, an imaginary number sqr2,

2 irrational transcendental constant numbers, e and pi,

and two irrational transcendental trigonometric variables, sine and cosines.

seemingly unrelated things are related

pi, sine, cosine are for circles or cycles

e is for natural or exponential growth or decay

i is used in electric current calculations

they are used in algebra, geometry, trigonometry, analysis, and electrical engineering

the Identity combines all 4 branches of mathematics and engineering in one equation

some mistakenly think e stands for exponent, it stands for Euler. It's known as Euler's number.

An interesting thing about the Identity is if you take natural logs, then

ln(e^ipi) = ln(-1)

ipi = the natural log of a negative number, which mathematicians say isn't possible

divide by i gives:

pi = ln(-1)/i?= about 3.141593?

and

i = ln(-1)/pi?= square root of -1?

just a few puzzles, brain teasers

e^ipi = e^3ipi= e^5ipi= e^7ipi=....= e^-ipi=e^-3ipi = e^-5ipi = e^-7ipi ...

= e^234,587ipi = e^-234,587ipi ....and infinitely more all = -1

e^2nipi = -1 where n= any integer.

It's a little too nippi outside, in most of the country today

Happy Holidays, Merry Christmas, Happy C(H)anukkah, Happy Kwanza, and for any other religious or atheist/agnostic days, Happy Something else y Prospero Nuevo Ano. As theoretical physicist Prof. Shelly Cooper noticed, Dec. 25 is Isaac Newton's birthday, although he needs a fire permit for the candles if he's still alive celebrating.