I have a question regarding the definition of integral for functions that are continuous in an open interval $(a, b)$ but are not necessarily defined or continuous at the endpoints $a, b$.
Courant and John, in the book "Introduction to Calculus and Analysis", define it in the following way:
If the limit $$F = \lim_\limits{\epsilon \to 0}
\int_{\alpha_\epsilon}^{\beta_\epsilon} f(x) \ dx$$ exists when $\
a < \alpha_\epsilon < \beta_\epsilon < b \ $ and $ \
\lim_\limits{\epsilon \to 0} \ \alpha_\epsilon = a, \ $ $
\lim_\limits{\epsilon \to 0} \ \beta_\epsilon = b,\ $ and if $F$ is
independent of the particular choice of $\alpha_\epsilon$ and
$\beta_\epsilon$ we say that the improper integral $\int_a^b f(x) \
dx$ converges and has value $F$.
The way I interpret this definition is:
- Let $(\alpha_n)_{n \in \mathbb{N}}$ and $(\beta_n)_{n \in \mathbb{N}}$ be two arbitrary real sequences converging to $a$ and $b$, respectively, such that $$(\forall n \in \mathbb{N}): \ a < \alpha_n < \beta_n < b. \ $$ If for each $n \in \mathbb{N},$ the definite integral $\int_{\alpha_n}^{\beta_n} f(x) \ dx $ has the same value $F$, we say that the improper integral $\int_a^b f(x) \ dx$ converges and has value $F$.
I feel they could have also wrote something like this:
- If the limit $$F = \lim_\limits{\epsilon \to 0} \int_{\alpha + \epsilon}^{\beta – \epsilon} f(x) \ dx$$ (where the points $\ \alpha + \epsilon\ $ and $ \ \beta – \epsilon \ $ approach the endpoints $a$ and $b$, respectively, from the interior of the interval) exists, then we say that the improper integral $\int_a^b f(x) \ dx$ converges and has value $F$.
Are these rewordings of the definition correct? I'm just not quite sure what they mean by "if $F$ is independent of the particular choice of $\alpha_\epsilon$ and $\beta_\epsilon$".
Best Answer
A few things to mention here.
The first definition says $\lim_{t \to 0} \alpha_\epsilon = \lim _{t \to 0} \beta_\epsilon= 0$. Surely you want these to converge to the endpoints $\alpha,\beta$? Not sure if this is a typo on your behalf or their behalf.
In your interpretation of the first definition, you talk about the definite integral having the "same value F" along these sequences - this is not the case. The value of the integral may change, we're just looking for what it converges to. (The idea is that we take ANY pair of sequences and look at what the integral tends to. If it's the same value in every case, the improper integral is said to converge to that value. If even one choice of sequences yields a different value, the improper integral doesn't converge.)
Your second definition is not the same - the first allows the limit to be taken at separate rates, which is important.
To elaborate on that last point, consider evaluating
$$\int_{-\pi/2}^{\pi/2} \tan x \, \text{d} x$$
which does not converge by the first definition, but does by the second.
Why? We have $\int_{\alpha_n}^{\beta_n} \tan x \, \text{d}x = \ln \frac{\cos \alpha_n}{\cos \beta_n}$. If we use the second definition, then we always have $\alpha_n = -\beta_n$ and so the integral is $0$ for every $n$, hence $0$ in the limit. However if $\beta_n$ and $\alpha_n$ are independent, we can let $\beta_n \to \pi / 2$ at a much faster rate than $\alpha_n \to -\pi/2$, so asymptotically $\ln \frac{\cos \alpha_n}{\cos \beta_n} \sim +\infty$. Similarly, if $\alpha_n$ converges faster, then $\ln \frac{\cos \alpha_n}{\cos \beta_n} \sim -\infty$.
Your proposed definition is in fact related to the Cauchy Principal Value, which you may be interested in looking at.