Integrals of Complex Functions

[Bashyboy]

Homework Statement

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. Homework Equations

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.

 

[BvU]

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.

 

[Bashyboy]

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 ?

Yes, that it what I figured the trouble to be.

On the other hand, would you have a problem "passing i through" for a sum ? And an integral is the limit of a summation.

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.

 

[BiGyElLoWhAt]

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.

 

[BvU]

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.