Definition:Improper Integral

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Definition

An improper integral is a definite integral over an interval which is not closed, that is, open or half open, and whose limits of integration are the end points of that interval.

When the end point is not actually in the interval, the conventional definition of the definite integral is not valid.

Therefore we use the technique of limits to specify the integral.


Note: In the below, in all cases the necessary limits must exist in order for the definition to hold.


Examples

Half Open Intervals

Open Above

Let $f$ be a real function which is continuous on the half open interval $\hointr a b$.

Then the improper integral of $f$ over $\hointr a b$ is defined as:

$\displaystyle \int_a^{\mathop \to b} \map f t \rd t := \lim_{\gamma \mathop \to b} \int_a^\gamma \map f t \rd t$


Open Below

Let $f$ be a real function which is continuous on the half open interval $\hointl a b$.

Then the improper integral of $f$ over $\hointl a b$ is defined as:

$\displaystyle \int_{\mathop \to a}^b \map f t \rd t := \lim_{\gamma \mathop \to a} \int_\gamma^b \map f t \rd t$


Open Intervals

Let $f$ be a real function which is continuous on the open interval $\openint a b$.

Then the improper integral of $f$ over $\openint a b$ is defined as:

$\displaystyle \int_{\mathop \to a}^{\mathop \to b} \map f t \rd t := \lim_{\gamma \mathop \to a} \int_\gamma^c \map f t \rd t + \lim_{\gamma \mathop \to b} \int_c^\gamma \map f t \rd t$

for some $c \in \openint a b$.


Unbounded Closed Intervals

Unbounded Above

Let $f$ be a real function which is continuous on the unbounded closed interval $\hointr a {+\infty}$.

Then the improper integral of $f$ over $\hointr a {+\infty}$ is defined as:

$\displaystyle \int_a^{\mathop \to + \infty} \map f t \rd t := \lim_{\gamma \mathop \to +\infty} \int_a^\gamma \map f t \rd t$


Unbounded Below

Let $f$ be a real function which is continuous on the unbounded closed interval $\hointl {-\infty} b$.

Then the improper integral of $f$ over $\hointl {-\infty} b$ is defined as:

$\displaystyle \int_{\mathop \to -\infty}^b \map f t \rd t := \lim_{\gamma \mathop \to -\infty} \int_\gamma^b \map f t \rd t$


Unbounded Open Intervals

The same techniques can be modified for unbounded open intervals in the forms $\openint a {+\infty}$ and $\openint {-\infty} b$:


Unbounded Above

Let $f$ be a real function which is continuous on the unbounded open interval $\openint a {+\infty}$.

Then the improper integral of $f$ over $\openint a {+\infty}$ is defined as:

$\displaystyle \int_{\mathop \to a}^{\mathop \to +\infty} \map f t \rd t := \lim_{\gamma \mathop \to a} \int_\gamma^c \map f t \rd t + \lim_{\gamma \mathop \to +\infty} \int_c^\gamma \map f t \rd t$

for some $c \in \openint a {+\infty}$.


Unbounded Below

Let $f$ be a real function which is continuous on the unbounded open interval $\openint {-\infty} b$.

Then the improper integral of $f$ over $\openint {-\infty} b$ is defined as:

$\displaystyle \int_{\mathop \to -\infty}^{\mathop \to b} \map f t \rd t := \lim_{\gamma \mathop \to -\infty} \int_\gamma^c \map f t \rd t + \lim_{\gamma \mathop \to b} \int_c^\gamma \map f t \rd t$

for some $c \in \openint {-\infty} b$.


A specific and important instance of this occurs when the interval in question is the set of all real numbers:


Unbounded Above and Below

Let $f$ be a real function which is continuous everywhere.

Then the improper integral of $f$ over $\R$ is defined as:

$\displaystyle \int_{\mathop \to -\infty}^{\mathop \to +\infty} \map f t \rd t := \lim_{\gamma \mathop \to -\infty} \int_\gamma^c \map f t \rd t + \lim_{\gamma \mathop \to +\infty} \int_c^\gamma \map f t \rd t$

for some $c \in \R$.

Usually $c$ is taken to be $0$ as this usually simplifies the evaluation of the expressions.


Notation

It is common practice to remove the $\to$ sign from the limits of integration, for example: $\displaystyle \int_{-\infty}^{+\infty} \map f t \rd t$.

However, this is not recommended, as confusion can result, in particular when investigating Lebesgue integration.


Sources