complex no. problem (1 Viewer)

[Faera]

Hey- not sure if this is in the syllabus any more, but:

Use the result (cos@ + isin@)^n = e^(i@) to prove DeMoivre's theorem, namely (cos@ + isin@)^n = cos(n@) + isin(n@).

 

[CM_Tutor]

1. e^i@ is not in the syllabus.

2. There is a typo in your question. It should read:

Use the result cos@ + isin@ = e^(i@) to prove ...

 

[Faera]

bah @ self. oops.
okay- thanks for that then.
yay! one less thing i need to know!

 

[:: ck ::]

thats eulers rule which was in the syllabus a long time ago (not anymore as cm tutor mentioned)

 

[CM_Tutor]

A pity, in a way, as it makes the proof of De Moivre's theorem two lines long.

 

[maniacguy]

While you're at it, prove Euler's rule. (This is trivial if I give you the only hint that springs to mind immediately, so I'm not giving you a hint. Your teacher may or may not have shown it to you)

 

[somedude2387]

help!!!

if w is a complex root of z^3 - 1 =0 where w does not equal to 1
show that 1 + w + w^2 = 0 (easy)
and hence evaluate
(1 - w)(1 - w^2)(1 - w^4)(1 - w^8)

help me out

 

[McLake]

Originally posted by maniacguy
While you're at it, prove Euler's rule. (This is trivial if I give you the only hint that springs to mind immediately, so I'm not giving you a hint. Your teacher may or may not have shown it to you)

I've seen a proof, it's very nice ...

 

[Grey Council]

ha! we hafta prove something which we know nothing about. ^__^

i've seen a proof of it too, but I payed zero attention to it. I'm struggling with normal 4u, thank you very much. I don't think I need extra rules and laws. heh

btw, if your interested in learning about it, its in the coroneos books. Coroneos 4u books ( the old ones ).

 

[Faera]

oooh, okay- thanks! Ive got those tucked away some where. i'll go take a look.

=)

 

[jonjon]

i wonder how ud graph e^(i@) ???

 

[Affinity]

somedude: notice w^4 = w and w^8 = w^2

maniacguy: 'trivial' hmm....

 

[CM_Tutor]

Originally posted by maniacguy
While you're at it, prove Euler's rule. (This is trivial if I give you the only hint that springs to mind immediately, so I'm not giving you a hint. Your teacher may or may not have shown it to you)

This was asked as question 8(c) on the Pymble Ladies' College paper of 1999. The question was worth 7 marks, and said:

--- Quote ---

Certain functions may be expressed as n infinite series called Taylor's Series using the equation

f(x) = f(0) + f'(0) * x +f''(0) * x² / 2! + ... + f^k(0) * x^k / k! + ...

Use this equation to find the Taylor Series for the functions
(i) f(x) = sin x
(ii) f(x) = cos x
(iii) f(x) = e^ix

Using these series, can you find a relationship between these three given functions?

Write down the value of e^i*pi

--- Quote ---

The question should really have added the facts that:
1. f^k(0) means the value of d^ky/dx^k, where y = f(x), at x = 0.
2. Calculus on complex functions follows the same rules as calculus on real functions, and so if f(x) = e^ix,
then f'(x) = i * e^ix.

 

[Giant Lobster]

damn... first time ive seen this formula. Mad ive learnt something

ok lets try do this:

sinx = x - x^3/3! + x^5/5! -x^7/7! + ... + ((-1)^n)*(x^(2n+1))/(2n+1)! as n ---> infinite
cosx = 1 - x^2/2! + x^4/4! - ... + ((-1)^n)*(x^2n)/(2n)! as n ---> infinite

And e is... ahhh Ive worked it out, but i cant be bothered writing it. Its just a combination of cos and sin series. And then u can take out i as a factor from the sin series, and you've just proven euler's thingy

hence cis@ = e^ix
hence e^ipi = cis pi = -1

whoa man my first q8! cool!

 

[CM_Tutor]

That's great. Now when you see truncated Taylor Series for sin x, cos x and (probably) e^x, you'll recognise them, like in 1994 4u HSC, Q7(a).

Note also that the 1997 4u HSC, Q 6(a) effectively established the Taylor Series for tan^-1x, as well as Gregory's series for pi / 4, which is:

pi / 4 = 1 - (1 / 3) + (1 / 5) - (1 / 7) + ...

 

[Giant Lobster]

hmmm actually thats the unnecessarily long proof for the euler thing. one can just differentiate and integrate 4 lines. "trivial" indeed.

 

[George W. Bush]

For proving Euler's therom, if I remember the general method of proof properly

Let z = cisx
z' = - sinx + icosx
= i(cisx)
i.e. z'/z = i
ln z = ix + c
c = 0
therefore ln z = ix
z = e^ix
cisx = e^ix

 

[CM_Tutor]

Originally posted by Giant Lobster
hmmm actually thats the unnecessarily long proof for the euler thing. one can just differentiate and integrate 4 lines. "trivial" indeed.

I never said it was the easiet proof ...