- #1
Wisc17
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Since learning about being able to complexify differential equations (I am doing the MIT OCW course by Arthur Mattuck), I have tried to apply to this to particular problems in integration as well. Specifically, I have tried to integrate the secant of some function to see if it would lead to the same accepted answer as when you used trignometric identities and u substitutions. What I have done is:
∫secΘdΘ = ∫(1/cosΘ)dΘ = Re{∫(1/eiΘ)dΘ}
∫(1/eiΘ)dΘ = ∫e-iΘdΘ = -eiΘ/i = -1/(i(cosΘ+isinΘ))
= -1/(-sin+icosΘ) = -1(-sin-icosΘ)/(sin2-(-cos2Θ)) = sinΘ+icosΘ
Re(sinΘ+icosΘ) = sinΘ
I know sinΘ is not the correct answer, but I do not understand why I cannot do the math this way. In the video I watched for complexifying integrals, I watched the professor do:
∫cosΘdΘ = Re{∫eiΘdΘ}
and the professor got the correct answer doing this, so I do not understand why I cannot do this with the secant.
I have limited access to teachers right now, but I did visit one teacher at a local college. He said I cannot complexify the equation as I have done, rather that I should have done this:
cosΘ = (eiΘ+e-iΘ)/2
However, he did not explain why, and before I could ask he went off and explained how to do the integral in the "traditional way." I am not interested in the traditional way; I am trying to find another way to do the integral, and am trying to find the hole in my knowledge regarding complexifying problems.
Any help would be appreciated. Thank you.
∫secΘdΘ = ∫(1/cosΘ)dΘ = Re{∫(1/eiΘ)dΘ}
∫(1/eiΘ)dΘ = ∫e-iΘdΘ = -eiΘ/i = -1/(i(cosΘ+isinΘ))
= -1/(-sin+icosΘ) = -1(-sin-icosΘ)/(sin2-(-cos2Θ)) = sinΘ+icosΘ
Re(sinΘ+icosΘ) = sinΘ
I know sinΘ is not the correct answer, but I do not understand why I cannot do the math this way. In the video I watched for complexifying integrals, I watched the professor do:
∫cosΘdΘ = Re{∫eiΘdΘ}
and the professor got the correct answer doing this, so I do not understand why I cannot do this with the secant.
I have limited access to teachers right now, but I did visit one teacher at a local college. He said I cannot complexify the equation as I have done, rather that I should have done this:
cosΘ = (eiΘ+e-iΘ)/2
However, he did not explain why, and before I could ask he went off and explained how to do the integral in the "traditional way." I am not interested in the traditional way; I am trying to find another way to do the integral, and am trying to find the hole in my knowledge regarding complexifying problems.
Any help would be appreciated. Thank you.