Euler's rule e^iQ = cosQ+ isinQ

[Learner]

Hi guys,
Current doing math and having problem understand the relationship between trig and euler's rule, how does one represent/convert to the other term?

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Any good ebooks here at eda download you would recommand?? I have tried the university library on this specific proof and topic and have yet to find a book with a clear and simple explaination, your help would be greatly appreciated!!!

Thanks!

 

[Sal]

Hi

http://en.wikipedia.org/wiki/Euler's_Formula

Sal

 

[Kral]

Learner,
I'm not sure what you're asking. If you want to prove Eulers rule. Write the Taylor series expansion for e^x, substituting i*Theta for x. group the real and imaginary terms, and you end up with Euler's rule in series expansion form.

If you are asking how to get cosx, sinx from Eulers rule:
e^-x = cos(-x) - isin(-x) =
e^x + e^(-x) = cosx + isinx + cos(-x) + isin(-x)
cos(-x) = cos(x)
sin(-x) = -sin(x)
e^x + e^(-x) = 2cosx
cosx = (e^x +e^(-x))/2
~
In a simiar manner, you can get the expresion for sinx from e^x - e^(-x)
Regards,
Kral

 

[thecreator]

i think for physical interpretation of the formula
you shoul read signals and systems(M.J.Roberts)

 

[slavko]

here is two nice books about complex analysis:

1. An Introduction to Complex Analysis for Engineers
Michael D. Alder
June 3, 1997
177 page

http://rapidshare.de/files/11012615/An_Introduction_to_Complex_Analysis_for_Engineers.rar

2. Complex Analysis by George Cain

http://www.math.gatech.edu/~cain/winter99/complex.html

hope you enjoy in learning :wink:

 

[Element_115]

It's series expansion>

Find the series for "e^x"
Then the series for "Cos(x)"
Then the series for "Sin(x)"

Let x = jω (or iω)

but remember j^2 = -1

and when you factor out everything you will see that the

Series of e^jω = Cos(ω) - jSin(ω)

Another cool thing is j^j = .20788... !

 

[whynot910]

if u want to prove it, u should use talor series.it is easy.

 

[cherrytart]

Needham (in Visual Complex Analysis) has a geometric explanation of Euler's formula.

This argument is also at:

h**p://www.cut-the-knot.org/Curriculum/Calculus/SineCosine.shtml

 

[xischaune]

If you want the proof of the formula... taylor series is the most easy to understand...
A collection of proofs (19 in number) is available at
https://www.ics.uci.edu/~eppstein/junkyard/euler/

 

[Learner]

If you want the proof of the formula... taylor series is the most easy to understand...

Any example of this on the net?

Thanks everyone!

 

[amiralib]

taylor series

you can visit this website for understandig the taylor series
https://mathworld.wolfram.com/TaylorSeries.html

 

[safwatonline]

Another cool thing is j^j = .20788... !

hmm that is new, how u get that ?! and does the 0.20788 represent something?

 

[Element_115]

Another cool thing is j^j = .20788... !

hmm that is new, how u get that ?! and does the 0.20788 represent something?

It's a little trick, but true none the less.

e^jω = Cos(ω) + jSin(ω) !---> Let ω=pi/2 ---> e^j(pi/2) = 0 + j(1)

--->raise both sides of the last EQ to the power of "j" and rearrange.

(1)j^j = e^(j*j(pi/2)) ----> j^j = e^((-1)pi/2) = 0.207879576350762

Cheers

 

[safwatonline]

well , hmmmmmmm ,, is it ok to say x^j >>> this seems strange

 

[Kral]

Learner,l
Also, visit hp://www.kith.org/logos/things/euler.html
.
Another curiosity:
√i = (√2)/2 + i(√2)/2, (-√2)/2 - i(√2)/2
Regards,
Kral

 

[smxx]

hi

exp(x) =1+ x/1! + (x²/2! )+.....x^n /n!+....

cos(x)=1-x^2+x^4/4!-x^6/6!+......
sin(x)=x-x^3/3!+x^5/5!-

exp(jw)=1+jw/1!-w^2/2!-jw^3/3!......
exp(-jw)=1-jw/1!-w^2/2!+jw^3/3!......

exp(jw)+exp(-jw)= 2-w^2/2!+w^4/4!...= 2cos(w)
exp(jw)-exp(-jw)=2jw/1!- ....= 2j sin(w)

therefor 2 exp(jw)= 2cos(w)+2jsin(w) ===>exp(jw)=cos(w)+jsin(w)