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Integrals of Complex Functions

  1. Nov 19, 2014 #1
    1. The problem statement, all variables and given/known data
    Suppose we have the function ##f : I \rightarrow \mathbb{C}##, where ##I## is some interval of ##\mathbb{R}## the functions can be written as ##f(t) = u_1(t) + i v(t)##. Furthermore, suppose this function is integral over the interval ##a \le t \le b##, which can be found by computing

    ##\int_a^b f(t) = \int_a^b u(t) + i \int_a^b v(t)##

    Let ##c## be some arbitrary complex constant. Proof that ## c \int_a^b f(t) = \int_a^b c f(t)##

    2. Relevant equations



    3. The attempt at a solution

    ## c \int_a^b f(t) = c \left[ \int_a^b u(t) + i \int_a^b v(t) \right] ##

    ## = c \int_a^b u(t) + ci \int_a^b v(t)##

    Let ##c = a + bi##,

    ##c \int_a^b f(t) = (a+bi) \int_a^b u(t) + (a+bi) i \int_a^b v(t)##

    ##= a \int_a^b u(t) + bi \int_a^b u(t) + ai \int_a^b v(t) - b \int_a^b v(t)##

    Here is where I had some trouble. The integrals of ##u(t)## and ##v(t)### are integrals of real-valued functions, and so I know that I can pass real-valued scalars through the integral sign; however, I do not know if I can pass ##i## through. I tried various manipulations, but all were positively unhelpful.
     
  2. jcsd
  3. Nov 19, 2014 #2

    BvU

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    Passing i through is what you are asked to prove (for the case c = i), so it becomes a bit circular then, doesn't it ?

    On the other hand, would you have a problem "passing i through" for a sum ? And an integral is the limit of a summation.
     
  4. Nov 19, 2014 #3
    Yes, that it what I figured the trouble to be.

    Hmmm, I am not sure. We have not yet viewed the integral as the limit of a sum yet, nor has the textbook represented it as such. There is obviously something elementary that I am missing.

    Here is the textbook that we are using: http://www.jiblm.org/downloads/jiblmjournal/V090515/V090515.pdf

    We are presently on chapter four, which begins on page 30. Perhaps you might see some theorem I am suppose to use. I will continue to scour through the text myself.
     
  5. Nov 19, 2014 #4

    BiGyElLoWhAt

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    Wow, there really isn't anything about riemann sums or anything...
    Anyways, there's a couple useful things (ones a definition and the other's a theorem) on the next couple pages. Try using those.
     
  6. Nov 20, 2014 #5

    BvU

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    Since, in the notes, it says "the proof of this theorem is very straightforward" I wouldn't worry about "passing through" the factor i.

    You could split in real part and imaginary part to verify.
     
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