Applications of complex numbers in other topics? (1 Viewer)

[~ ReNcH ~]

Is the use of complex numbers confined to that topic, or is it necessary to introduce complex numbers into other 4U topics as well? E.g. do you need to use complex numbers to solve Mechanics, Integration... questions, or is it independent of the other topics?

 

[SeDaTeD]

Complex numbers would only be involve with polynomials, for 4unit. It shouldn.t be involved in mechanics, integration etc. However it could be possible that there may be locus questions which are related to conics, but that's not common.

 

[dawso]

ok, we used complex in deriving stuff for conics (rectangular hyperbola), and mechanics, and in questions in polynomials, um, cant think of anything else, oh, u sometimes get inequality (harder 3u) with complex numbers included....

 

[Affinity]

solving y'' + y = 0 (and the like)could involve complex numbers

 

[redslert]

solving y'' + y = 0 (and the like)could involve complex numbers

no need to scare them with uni maths so early!

 

[~ ReNcH ~]

Rightio

Suppose you had to integrate something like sin⁵@. Would you have to use De Moivre's theorem to convert it into an integratable form? (in which case complex numbers would come into Integration). Or is there another easier method that I haven't learnt yet?

 

[McLake]

Rightio

Suppose you had to integrate something like sin⁵@. Would you have to use De Moivre's theorem to convert it into an integratable form? (in which case complex numbers would come into Integration). Or is there another easier method that I haven't learnt yet?

You can use De Moivre's theorem, or you can do the great split trick:

sin⁵@
= sin@ * sin⁴@
= sin@ * (sin²@)²
= sin@ * (1 - cos²@)²
= sin@ * (1 - 2 * cos²@ + cos⁴@)
= sin@ - 2 * sin@ * cos²@ + sin@ * cos⁴@

now this is easy to intergrate (since we have the form I (f'(x) * f(x)) dx = 1/2f(x)²)

I = -cos@ + 2/3 * cos³@ - 1/5 * cos⁵@ + C

EDIT: Some constants would be nice ...

 

[~ ReNcH ~]

You can use De Moivre's theorem, or you can do the great split trick:

sin⁵@
= sin@ * sin⁴@
= sin@ * (sin²@)²
= sin@ * (1 - cos²@)²
= sin@ * (1 - 2 * cos²@ + cos⁴@)
= sin@ - 2 * sin@ * cos²@ + sin@ * cos⁴@

now this is easy to intergrate (since we have the form I (f'(x) * f(x)) dx = 1/2f(x)²)

I = -cos@ + 2/3 * cos³@ - 1/5 * cos⁵@ + C

EDIT: Some constants would be nice ...

Ok....I haven't really learnt that method of integration (i.e. splitting), but I read over it briefly from Cambridge. I'll probably stick to your method...de Moivre's is too long.

 

[lucifel]

and after all that you realise you could just use substitution,

'put u = cos x
du/dx = sin
1-u = sinx
(1-u) = sinx

so now your integration becomes:

(1-u)du
=(1- 2u + u) du (which is easy to integrate)

come to think of it, its the same method, only it looks nicer with 'u's instead of sin x (neater, kind of...ok i am rambling..)