e^(i*theta)= cis(theta) (1 Viewer)

[kaz1]

Just wondering how mathematicians came up with this cause you can't multiply a number an imaginary number of times (at least I think so). Is there a proof or is it just an axiom or something?

 

[tommykins]

e^x = 1+x+x^2/2! + x^3/3! + x^4/4! .....x^n/n!

replace x with ix

e^ix = 1+ix-x^2/2!+ix^3/3!...
Then by collecting the real and imaginary terms, you get (expression 1) + i(expression 2)

it just so happens that expression one equates to cos(x) which is 1+x^2/2!-x^4/4! + x^6/6! etc. etc. and the expression 2 equates to sinx

hence e^ix = cosx + isinx

PS. it'll become more clear when you study taylor series

 

[Cazic]

e^x = 1+x+x^2/2! + x^3/3! + x^4/4! .....x^n/n! ...

Fixed.

There is also a nice proof on Wikipedia that only uses differentiation iirc. I'm wondering if the OP had the same concern when they were multiplying negative numbers? "You can't multiply a number a negative number of times ..."

 

[untouchablecuz]

There is a nice proof on Wikipedia that only uses differentiation. I'm wondering if the OP had the same concern when they were multiplying negative numbers? "You can't multiply a number a negative number of times ..."

but what does it actually "mean" to differentiate a complex number? how do we justify this?

(just raising some q's for discussion)

 

[shaon0]

Well dz/d@=iz
dz/z=i d@
ln(z)+C=i@
ln(z)=i@ as C=0
z=e^i@

 

[Gussy Booo]

Nop, not an Axiom. There are two ways of resolving this. However, one way consists of very advanced algebra...I THINK.

I much prefer to accept this geometrically. For I am still a H.S. student.
You basically differentiate the Graph , and you'll notice something very very intresting.
Here:

Question Corner -- Why is e^(pi*i) = -1?

 

[Cazic]

but what does it actually "mean" to differentiate a complex number? how do we justify this?

(just raising some q's for discussion)

The derivative of a complex valued function is defined similarly to the real valued case: Let f be a real valued function. The the derivative of f at the point z₀ is given by:



The only subtle difference is that z must be allowed to approach z₀ from any direction. Recall, in the real case a limit can approach from the left or right, whereas in the complex plane the limit must exist when approaching from any direction. I can flesh this out a bit more if you need, since I don't think left and right hand limits are covered in the HSC (though they should be).

The derivative of any complex number (say i) is still 0. Intuitively you would expect this since a constant is ... well, constant.

 

[untouchablecuz]

The derivative of a complex valued function is defined similarly to the real valued case: Let f be a real valued function. The the derivative of f at the point z₀ is given by:



The only subtle difference is that the limit must be allowed to approach z₀ from any direction. Recall, in the real case a limit can approach from the left or right, whereas in the complex plane the limit must exist when approaching from any direction. I can flesh this out a bit more if you need, since I don't think left and right hand limits are covered in the HSC (though they should be).

The derivative of any complex number (say i) is still 0. Intuitively you would expect this since a constant is ... well, constant.

nah its alright, covered left/right hand limits in my uni math course

how is the derivative interpreted in the complex plane? e.g. it is the gradient function for real valued functions, how so for complex valued functions?

(sorry about being annoying, ive read a few pdfs on complex analysis and they havent really explained this)

 

[Cazic]

It's hard to give any geometric intuition for the complex derivative since C is a 2-dimensional (real) vector space, so we have 4 real dimensions to try and interpret. My own interpretation is: at the very least, it's a measure of change - but I split the change into two components. The complex derivative gives a instantaneous rate of "magnitude" change and "angle" change. For example, if you calculate some derivative at a point and it's i, the instantaneous rate of change of the function is a change in angle of pi/2. The intuition is a little harder to build I guess, but it's doable.

 

[Rezen]

From Coroneos,