renegade05
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- Sep 10, 2010
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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
. . . . .\(\displaystyle \displaystyle{\left|\int_a^b\, f(t)\, dt\right|\, \leq\, \int_a^b\, |f(t)|\, dt\, \leq\, C(b\, -\, a)}\)
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?
. . . . .\(\displaystyle \displaystyle{\int_a^b\, f(t)\, dt\, =\, \int_a^b\, u(t)\, dt\, +\, i\int_a^b\, v(t)\, dt}\)
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
. . . . .\(\displaystyle \displaystyle{\left|\int_a^b\, f(t)\, dt\right|\, \leq\, \int_a^b\, |f(t)|\, dt\, \leq\, C(b\, -\, a)}\)
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?
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