Using complex numbers for evaluating integrals

In summary, to evaluate the integral from 0 to pi { e^2x cos 4x dx }, you need to take the real part of the second integral from 0 to pi { e^((2 + 4i)x) dx}.
  • #1
genjix
3
0
How can I use complex numbers to evaluate an integral? For instance I'm reading a book on complex numbers and it says that to evaluate the integral from 0 to pi { e^2x cos 4x dx }, I must take the real part of the integral from 0 to pi { e^((2 + 4i)x) dx}.

It totally skips how you do that. I don't see how taking the real part of the second integral relates to the first one. If I try to integrate e^((2 + 4i)x) and then take the real part, I eventually get:

Re(z) = 1/5 e^2x (1/2 cos 4x + sin 4x)

But I don't see any relation whatsoever, or how they did this.
 
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  • #2
welcome to pf!

hi genjix! welcome to pf! :smile:

(try using the X2 button just above the Reply box :wink:)
genjix said:
If I try to integrate e^((2 + 4i)x) and then take the real part, I eventually get:

Re(z) = 1/5 e^2x (1/2 cos 4x + sin 4x)

show us how you did this :smile:
 
  • #3
Well I went onto #math on Freenode and someone gave me the missing puzzle piece. Basically that Re(integral f(x)) = integral Re(f(x)). With that in hand, everything is obvious:

Take the real part of e^2x e^4ix
Re(e^2x e^4ix) = e^2x (cos 4x + i sin 4x) ... from de moivre and because the Re(integral f(x)) = integral Re(f(x)), we can say that Re(integral e^2x e^4ix) = integral Re(e^2x e^4ix) = integral Re(e^2x (cos 4x + i sin 4x)) = integral e^2x cos 4x

now the integral e^2x e^4ix = integral e^(2 + 4i)x and to integrate exponentials we have the usual integral e^cx = 1/c e^cx

so we get:
1/(2 + 4i) e^{(2 + 4i)x}
http://mathbin.net/110072 [Broken] (multiply top and bottom of fraction by (2 - 4i) on first line)
now we have the bottom line to be evaluated from 0 to +pi (as per the original integral)
cos 4pi = cos 0 = 1
sin 4pi = sin 0 = 0

so we get the final line as being:
1/5 e^2pi (1/2 cos 4pi + sin 4 pi) - 1/5 e^0 (1/2 cos 0 + sin 0) = 1/10 (e^2 - 1)
 
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  • #4
I just did, but my post isn't showing up? I try to click back in my browser and submit it, but it is saying I've made a duplicate post strangely.

I found someone on #math in Freenode who told me that the Re(integral f(x)) = integral Re(f(x)) which is the missing puzzle piece I needed to work everything out. The post I've tried to make here shows all my working out for others to see how it is done.
 
  • #5
hi genjix! :smile:

(just got up :zzz:)
genjix said:
Re(integral f(x)) = integral Re(f(x))

yes, that's right :smile:

(if you're still unsure, try submitting your proof again)
 

1. What are complex numbers and why are they used for evaluating integrals?

Complex numbers are numbers that are composed of both a real part and an imaginary part. They are used in mathematics and science to represent quantities that cannot be expressed using only real numbers. Complex numbers are used for evaluating integrals because they allow for the integration of functions that involve complex quantities, making it possible to solve a wider range of integrals.

2. How are complex numbers used in the process of evaluating integrals?

Complex numbers are used in the process of evaluating integrals by allowing for the integration of functions that involve complex quantities. This is done by converting the integral into a complex form and then using various techniques, such as the Cauchy-Riemann equations, to solve the integral.

3. What are some common techniques for using complex numbers to evaluate integrals?

Some common techniques for using complex numbers to evaluate integrals include converting the integral into a complex form, using the Cauchy-Riemann equations, using contour integration, and using the residue theorem. These techniques allow for the integration of functions involving complex quantities, making it possible to solve a wider range of integrals.

4. Can complex numbers be used to evaluate any type of integral?

No, complex numbers cannot be used to evaluate any type of integral. They are most commonly used to evaluate integrals involving complex quantities, such as complex-valued functions. They may also be used in certain cases to simplify the evaluation of real-valued integrals.

5. Are there any limitations to using complex numbers for evaluating integrals?

Yes, there are limitations to using complex numbers for evaluating integrals. One limitation is that the techniques used for evaluating complex integrals can be complex and require a strong understanding of complex analysis. Additionally, not all integrals can be solved using complex numbers, so it is important to understand when and how to use them effectively.

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