- #1
Hoplite
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When is it alright to bring the limit within the integral?
In other words, when is it true to say...
lim∫f(x)dx = ∫limf(x)dx
?
In other words, when is it true to say...
lim∫f(x)dx = ∫limf(x)dx
?
dextercioby said:Did u mean
[tex] \substack{\displaystyle{!}\\ \displaystyle{=}} [/tex]...? (click on the code)
Daniel.
Hoplite said:What I mean is:
[tex]\lim_{k\rightarrow\infty} \int f_{k}(z)dz = \int\lim_{k\rightarrow\infty}f_{k}(z)dz[/tex]
In this case, [tex]f_{k}(z)[/tex] is the zeta function,
[tex]f_{k}(z) = \sum^k_{n=1}\frac{1}{n^z}[/tex]
The general rule is that if the integrand is continuous and the limits of integration are finite, then the limit can be brought inside the integral.
No, the limit cannot be brought inside the integral if the integrand is discontinuous. In such cases, the integral must be evaluated using methods such as the Cauchy principal value.
Yes, there are some exceptions to the rule. For example, if the integral is improper, the limit may need to be evaluated separately before bringing it inside the integral. Also, if the integrand has an essential singularity at one of the limits of integration, the limit cannot be brought inside the integral.
Bringing the limit inside the integral does not change the value of the integral as long as the general rule is followed. This is because the limit is evaluated after the integration has been performed.
Bringing the limit inside the integral can be useful when dealing with certain types of integrals, such as those involving infinite limits or integrands with many terms. It can also simplify the integration process by allowing us to use known integration rules and techniques.