Integrating Complex Numbers (1 Viewer)

[Gussy Booo]

Is there anything wrong with integrating complex numbers? I've heard that some teachers/lecturers get frustrated when they see students attempting to integrate complex numbers. Can anyone explain why this may be bad. All I know is this :

""Yes, in practical terms integration of complex-valued functions is no different from integration of real-valued functions: the Fundamental Theorem of Calculus is true for both (although the Mean Value Theorem fails for complex-valued functions).""

 

[Affinity]

If you are talking about complex valued functions (f:R --> C) then it's relatively straight forward because you can consider f(x) = g(x) + i*h(x) where g, h are regular real functions.

What gets interesting is when you consider complex functions ie f:C --> C.

for one where a and b are two complex numbers is not well defined anymore and you will have to consider specific paths between a and b. so things work slightly differently in this case. ofcourse there's the cauchy integral formula which makes integrals path independent for nice functions

 

[MetroMattums]

It isn't in the syllabus and teachers aren't paid to teach that stuff. If you start asking questions about it then they have a mental breakdown..

 

[Gussy Booo]

I see.....
It's just that, I saw an alternative way in solving integrals such as [Int] e^xcosx dx
It consists of integrating Eulers formula (Which is e^ix=cosx+isinx).
By doing this, you then equate real and imaginary parts, and I think you save yourself a heck a lot of time.

 

[MetroMattums]

Yeah that would save you a lot of effort.

If you did that in an HSC exam the markers shouldn't mark you down - it is 4U after all - unless they're complete assholes.

 

[tommykins]

I see.....
It's just that, I saw an alternative way in solving integrals such as [Int] e^xcosx dx
It consists of integrating Eulers formula (Which is e^ix=cosx+isinx).
By doing this, you then equate real and imaginary parts, and I think you save yourself a heck a lot of time.

how would you use eulers to solve that integral anywyas?

i can't see any obvious implication

 

[tacogym27101990]

how would you use eulers to solve that integral anywyas?

i can't see any obvious implication

S(e^xcosx)dx = Re[S e^x*e^ixdx]
=Re[Se^((1+i)x) dx]
=e^xRe[((1-i)/2)e^(i)x dx] + C
=(1/2)e^x(cosx + sinx) +C

I guess its a bit quicker than recurrence integrals

 

[Gussy Booo]

S(e^xcosx)dx = Re[S e^x*e^ixdx]
=Re[Se^((1+i)x) dx]
=e^xRe[((1-i)/2)e^(i)x dx] + C
=(1/2)e^x(cosx + sinx) +C

I guess its a bit quicker than recurrence integrals

Yes. Something like that .

If it even saves me 1 minute. I'll take it.

 

[shaon0]

Yes. Something like that .

If it even saves me 1 minute. I'll take it.

I wouldn't risk losing 2 marks...the markers are testing your knowledge on IBP not how to use Euler's formula. You'll probably get 0 marks for that in the HSC unless you rigorously prove Euler's formula.

 

[Gussy Booo]

I wouldn't risk losing 2 marks...the markers are testing your knowledge on IBP not how to use Euler's formula. You'll probably get 0 marks for that in the HSC unless you rigorously prove Euler's formula.

Point taken.

 

[nikkifc]

I wouldn't risk losing 2 marks...the markers are testing your knowledge on IBP not how to use Euler's formula. You'll probably get 0 marks for that in the HSC unless you rigorously prove Euler's formula.

Do you prove the fundamental theorem of calculus everytime you calculate an integral?

 

[Trebla]

Oh god here we go again....

 

[shaon0]

Oh god here we go again....

My sentiments exactly. I'm not going to debate the fact (once again) that for HSC Maths, one does not require tertiary level mathematics. That's why it is HSC, not Math3031.