complex valued / riemann sums

[renegade05]

3) For a complex-valued function $\displaystyle f(t)\, =\, u(t)\, +\, iv(t),$ where $\displaystyle u$ and $\displaystyle v$ are integrable real functions, the integral of $\displaystyle f$ over an interval $\displaystyle [a,\, b]$ of the real line is given by:

. . . . .$\displaystyle \displaystyle{\int_a^b\, f(t)\, dt\, =\, \int_a^b\, u(t)\, dt\, +\, i\int_a^b\, v(t)\, dt}$

Show that one can define the integral by Riemann sums also for complex-valued functions; i.e., show that

. . . . .$\displaystyle \displaystyle{\int_a^b\, f(x)\, dt\, =\, \lim_{n\, \to\, \infty}\,\sum_{k=1}^n\, f\left(a\, +\, k\frac{b\, -\, a}{n}\right)\, \frac{b\, -\, a}{n}}$

You may use the fact that Riemann sums converge to the integral for real-valued functions.

4) Let $\displaystyle f(t)$ be complex-valued and integrable over $\displaystyle [a,\, b]\, (a\, <\, b)$ and let $\displaystyle |f(t)|\, \leq\, C$ for some $\displaystyle C\, >\, 0.$ Using the definition as limit of Riemann sums as above, show that

. . . . .$\displaystyle \displaystyle{\left|\int_a^b\, f(t)\, dt\right|\, \leq\, \int_a^b\, |f(t)|\, dt\, \leq\, C(b\, -\, a)}$

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These are the problems I am working with.

So I am having problems with both. I'm not sure how to apply the definition of the Riemann sum to the complex integral? or really answer the question.

How can I start this proof off?

And number 4 is also giving me a hard time. How can I show the equality holds using the definition as limit of riemann sums?

 

[HallsofIvy]

You are told, by $\displaystyle \int_a^b f(t)dt= \int_a^b u(t)dt+ i\int_a^b v(t)dt$, that the Riemann integral of a complex valued function is just the sum of two Riemann integrals of real valued functions. Just apply the definition of "Riemann Integral" to both.

 

[renegade05]

You are told, by $\displaystyle \int_a^b f(t)dt= \int_a^b u(t)dt+ i\int_a^b v(t)dt$, that the Riemann integral of a complex valued function is just the sum of two Riemann integrals of real valued functions. Just apply the definition of "Riemann Integral" to both.

I'm really sorry. I know this is an easy problem I just don't know where to apply the Riemann integral. I feel like the question answers itself and I'm not how to answer this question with adding new information.

Please some more guidance.. and on question 4 as well ...

Thanks!

 

[daon2]

I'm really sorry. I know this is an easy problem I just don't know where to apply the Riemann integral. I feel like the question answers itself and I'm not how to answer this question with adding new information.

Please some more guidance.. and on question 4 as well ...

Thanks!

Let $\displaystyle \Delta x = (b-a)/n$

$\displaystyle \displaystyle \int_a^b u(t) dt = \lim_{n\to\infty} \sum_{k=1}^n u(a+k\Delta x)\cdot \Delta x$

$\displaystyle \displaystyle \int_a^b v(t) dt = \lim_{n\to\infty} \sum_{k=1}^n v(a+k\Delta x)\cdot \Delta x$

$\displaystyle \displaystyle \int_a^b u(t) dt + i\int_a^b v(t) dt = \lim_{n\to\infty} \sum_{k=1}^n u(a+k\Delta x)\cdot \Delta x + i\cdot \lim_{n\to\infty} \sum_{k=1}^n v(a+k\Delta x)\cdot \Delta x$

$\displaystyle \displaystyle = \lim_{n\to\infty} \sum_{k=1}^n \left( u(a+k\Delta x) + i\cdot v(a+k\Delta x) \right)\Delta x = ?$



For 4, use the triangle inequality:

$\displaystyle \left|\sum a_k \right| \le \sum |a_k|$

and the fact that the limit can move inside/outside absolute values (absolute value is a continuous function).