Improper Integrals

13.1 Preliminaries

Definition 13.1.1: Let $I$ be an interval of the form $[a,\infty)$, $(-\infty,b]$, or $(-\infty,\infty)$ nad let $f$ be a function defined on $I$. We say that $f$ is integrable on $I$ if $f$ is integrable on each compact interval contained in $I$. The integral of $f$ over $I$ is said to be of the first kind. In each case, the integral of $f$ over $I$ is defined as follows: 

a) If $I=[a,\infty)$, then $\int_a^\infty f(x)dx=\lim_{b\rightarrow \infty} \int_a^b f(x)dx$

b) If $I=(-\infty, b]$, then $\int_{-\infty}^b f(x)dx=\lim_{a\rightarrow -\infty} \int_a^b f(x)dx$

c) If $I=(-\infty,\infty)$, then $\int_{-\infty}^\infty f(x)dx=\lim_{b\rightarrow \infty,\ a\rightarrow -\infty} \int_a^b f(x)dx$

If the limit exists and has value $L$, then the improper integral of $f$ over $I$ is said to converge to $L$. If the limit fails to exist, then the improper integral is said to diverge.

Definition 13.1.2: Let $I=(a,b]$ and let $f$ be a real-valued function defined on $I$. Suppose also that $f$ has a pole at the endpoint $a$ and that, for each $c$ in $(a,b)$, $f$ is bounded on $[c,b]$. Then $f$ is said to be integrable on $I$ if $f$ is integrable on $[c,d]$ for each $c$ in $(a,b]$. The improper integral of $f$ over $I$ is defined to be $$\int_a^b f(x)dx - \lim_{c\rightarrow a^+} \in_c^b f(x)dx$$ and is said to be of the second kind. If this limit exists and has value $L$, then the integral is said to converge to $L$. If the limit fails to exist, then the integral is said to diverge.

13.2 Improper Integrals of the First Kind

Theorem 13.2.1: If $f_1$ and $f_2$ are integrable functions on $[a,\infty)$ such that $\int_a^\infty f_1(x) dx$ and $\int_a^\infty f_2(x)dx$ converge and if $c_1$ and $c_2$ are real numbers, then $\int_a^\infty [c_1 f_1(x)+c_2f_2(x)]dx$ also converges and $$\int_a^\infty [c_1f_1(x) + c_2f_2(x)]dx=c_1\int_a^\infty f_1(x)dx + c_2 \int_a^\infty f_2(x)dx$$

Theorem 13.2.2: Suppose that the integrand $f$ is nonnegative and integrable on $[a,\infty)$. The integral $\int_a^\infty f(x)dx$ converges if and only if the set $S = \{ \int_a^b f(x)dx:\ b\mbox{ in }(a,\infty)\}$ is bounded above. In this event, $\int_a^\infty f(x)dx = \sup S$.

Theorem 13.2.3: Comparison Test; Integrals of First Kind: Let $f$ and $g$ be nonnegative, integrable functions on $[a,\infty)$. Suppose that there exists a $c>a$ such that $f(x)\le g(x)$ for all $x$ in $[c,\infty)$.

i) If $\int_a^\infty g(x) dx$ converges, then $\int_a^\infty f(x) dx$ also converges.

ii) If $\int_a^\infty f(x) dx$ diverges, then $\int_a^\infty g(x) dx$ also diverges.

Theorem 13.2.4 Limit Comparison Test; Positive Integrands: Let $f$ and $g$ be positive functions that are integrable on $[a,\infty)$. If $\lim_{x\rightarrow \infty} f(x)/g(x)=L$ exists and is positive, then $\int_a^\infty f(x)dx$ converges if and only if $\int_a^\infty g(x)dx$ converges.

Corollary 13.2.5: Let $f$ and $g$ be positive functions that are integrable on $[a,\infty)$.

i) If $\lim_{x\rightarrow \infty} f(x)/g(x) = 0$ and if $\int_a^\infty g(x)dx$ converges, then $\int_a^\infty f(x)dx$ converges.

ii) If $\lim_{x\rightarrow \infty} f(x)/g(x) = \infty$ and if $\int_a^\infty g(x)dx$ diverges, then $\int_a^\infty f(x)dx$ diverges.

Definition 13.2.1: Let $\int_a^\infty f(x)dx$ be an improper integral of the first kind.

i) If $\int_a^\infty \left| f(x)\right| dx$ converges, then $\int_a^\infty f(x)dx$ is said to converge absolutely.

ii) If $\int_a^\infty f(x) dx$ converges, but $\int_a^\infty \left| f(x)\right| dx$ diverges, then $\int_a^\infty f(x)dx$ is said to converge conditionally.

As with series so here we have following theorem.

Theorem 13.2.6: If $\int_a^\infty \left| f(x)\right| dx$ converges, then $\int_a^\infty f(x)dx$ also converges.

Theorem 13.2.7 Dirichlet's Test: Suppose that $\int_a^\infty f(x)g(x)dx$ is an improper integral of the first kind. Suppose also that $f$ and $g$ satisfy the following conditions: 

i) The function $f$ is continuous on $[a,\infty)$.

ii) The function $F(x)=\int_a^x f(t)dt$ is bounded on $[a,\infty)$.

iii) The function $g$ is differentiable on $[a,\infty)$, $g' \le 0$ and \lim_{x\rightarrow \infty} g(x)=0$

Then $\int_a^\infty f(x)g(x)dx$ converges.

Definition 13.2.2: Let $f$ be an integrable function on $\mathbb{R}$. The Cauchy principle value of $\int_{-\infty}^\infty f(x)dx$ is $\lim_{a\rightarrow \infty} \int_{-a}^a f(x)dx$, provided this limit exists.

Theorem 13.2.8: If $\int_{-\infty}^\infty f(x)dx$ converges to $L$, then the Cauchy principle value of the integral is $L$.

13.3 Improper Integrals of the Second Kind

Theorem 13.3.1: Suppose that the integrand $f$ is nonnegative and integrable on $(a,b]$ and has a pole at $x=a$. The improper integral $\int_a^b f(x)dx$ converges if and only f the set $S=\{\int_c^b f(x)dx:\ c\mbox{ in }(a,b)\}$ is bounded above. In this case, $\int_a^b f(x)dx= \sup S$

Theorem 13.3.2 Comparison Test; Integrals of Second Kind: Let $f$ and $g$ be nonnegative, integrable functions on $(a,b]$, each having a pole at $x=a$ such that, for some $c_0$, we have $f(x)\le g(x)$ for all $x$ in $(a,c_0]$.

i) If $\int_a^b g(x)dx$ converges, then $\int_a^b f(x) dx$ also converges.

ii) If $\int_a^b f(x)dx$ converges, then $\int_a^b g(x) dx$ also diverges.

We also have a limit comparison test for integrals of the second kind.

Theorem 13.3.3 Limit Comparison Test; Positive Integrands: Let $f$ and $g$ be positive, integrable functions on $(a,b]$, each having a pole at $x=a$. If $\lim_{x\rightarrow a^+} f(x)/g(x)=L$ exists and is positive, then $\int_a^b f(x)dx$ converges if and only if $\int_a^b g(x)dx$ converges.

Corollary 13.3.4: Let $f$ and $g$ be positive integrable functions on $(a,b]$, each having a pole at $x=a$.

i) If $\lim_{x\rightarrow a^+} f(x)/g(x)=0$ and if $\int_a^b g(x)dx$ converges, then $\int_a^b f(x)dx$ converges.

ii) If $\lim_{x\rightarrow a^+} f(x)/g(x)=\infty$ and if $\int_a^b g(x)dx$ diverges, then $\int_a^b f(x)dx$ diverges.

Definition 13.3.1: Let $\int_a^b f(x)dx$ be an improper integral of the second kind.

i) If $\int_a^b \left| f(x)\right| dx$ converges, then $\int_a^b f(x)dx$ is said to converge absolutely.

ii) If $\int_a^b f(x)dx$ converges but $\int_a^b \left| f(x)\right| dx$ diverges, then $\int_a^b f(x)dx$ is said to converge conditionally.

The following theorem is, of course, immediate.

Theorem 13.3.5: If $\int_a^b \left| f(x)\right| dx$ converges, then $\int_a^b f(x)dx$ also converges.

13.4 Uniform Convergence of Improper Integrals

Definition 13.4.1: Let $S$ be a nonempty subset of $\mathbb{R}$ and let $f$ be a real-valued function defined on $[a,\infty) \times S$ in $\mathbb{R}^2$. Suppose that, for each $x_2$ in $S$. $f(t,x_2)$ is integrable on $[a,\infty)$ and that $F(x_2)=\int_a^\infty f(t,x_2)dt$ is a convergent improper integral of the first kind. We say that $\int_a^\infty f(t,x_2)dt$ converges uniformly to $F$ on $S$ if, for every $\epsilon > 0$, there exists a $b_0 > a$ such that, for all $b\ge b_0$ and all $x_2$ in $S$, $$\left| F(x_2) - \int_a^b f(t,x_2)dt\right| < \epsilon$$ We write $\int_a^\infty f(t,x_2)dt=F(x_2)$ [uniformly] on $S$.

Definition 13.4.2: Let $S$ be a nonempty subset of $\mathbb{R}$ and let $f$ be a real-valued function defined on $(a,b]\times S$ in $\mathbb{R}^2$. Suppose that, for each $x_2$ in $S$, $f(t,x_2)$ has a pole at $t=a$ and is integrable on $(a,b]$, and that $F(x_2)=\int_a^b f(t,x_2)dt$ is a convergent, improper integral of the second kind. We say that $\int_a^b f(t,x_2)dt$ converges uniformly to $F$ on $S$ if, for each $\epsilon > 0$, there exists a $c_0$ in $(a,b)$ such that, for all $c$ in $(a,c_0)$ and for all $x_2$ in $S$, $$\left| F(x_2) - \int_a^b f(t,x_2)dt\right| < \epsilon$$ We write $\int_a^b f(t,x_2)dt = F(x_2)$ [uniformly] on $S$.

Theorem 13.4.1 Weierstrass's $M$-test; Integrals of the First Kind: Let $S$ be a nonempty subset of $\mathbb{R}$ and let $f$ be a real-valued function defined on $[a,\infty)\times S$ in $\mathbb{R}^2$. Suppose that, for each $x_2$ in $S$, $f(t,x_2)$ is integrable on $[a,\infty)$. Suppose also that there exists and integrable function $M(t)$ on $[a, \infty)$ with the following properties:

i) For all $(t,x_2)$ in $[a,\infty) \times S$, $\left| f(t, x_2)\right| \le M(t)$.

ii) $\int_a^\infty M(t) dt$ converges.

That $\int_a f(t,x_2)dt$ converges uniformly on $S$.

Theorem 13.4.2 Weierstrass's $M$-test; Integrals of the Second Kind: Let $S$ be a nonempty subset of $\mathbb{R}$ and let $f$ be a real-valued function defined on $(a,b]\times S$ in $\mathbb{R}^2$. Suppose that, for each $x_2$ in $S$, $f(t,x_2)$ has a pole at $a$ and is integrable on $(a,b]$. Suppose also that there exists an integrable function $M(t)$ on $(a,b]$ with the following properties:

i) For each $(t,x_2)$ in $(a,b]\times S$, $\left| f(t,x_2)\right| \le M(t)$

ii) $\int_a^b M(t)dt$ converges.

Then $\int_a^b f(t,x_2)dt$ converges uniformly on $S$.

Theorem 13.4.3: Let $S$ be a nonempty subset of $\mathbb{R}$ and let $f$ be a continuous, real-valued function on $[a,\infty)\times S$ in $\mathbb{R}^2$. Assume that $\int_a^\infty f(t,x_2)dt = F(x_2)$ [uniformly] on $S$. Then $F$ is continuous on $S$.

Theorem 13.4.4: Let $S$ be a nonempty subset of $\mathbb{R}$ and let $f$ be a continuous, real-valued function on $(a,b]\times S$ in $\mathbb{R}^2$. Suppose that for each $x_2$ in $S$. $f(t,x_2)$ has a pole at $t=a$ and is integrable on $(a,b]$. Suppose also that the improper integral $\int_a^b f(t,x_2)dt=F(x_2)$ [uniformly] on $S$. Then $F$ is continuous on $S$.

Theorem 13.4.5: Let $f$ be a real-valued function defined on $[a,\infty)\times [c,d]$ in $\mathbb{R}^2$ such that, for each $x_2$ in $[c,d]$, the integral $\int_a^\infty f(t,x_2)dt$ converges pointwise to a function $F$ on $[c,d]$. Suppose that the partial derivative $D_2f$ exists and is continuous on $[a,\infty)\times [c,d]$. Suppose further that the improper integral $\int_a^\infty D_2 f(t,x_2)dt$ converges uniformly on $[c,d]$. Then $F$ is differentiable on $[c,d]$ and $$F'(x_2)=\int_a^\infty D_2 f(t,x_2)dt$$

Theorem 13.4.6: Let $f$ be a real-valued function on $(a,b]\times [c,d]$ in $\mathbb{R}^2$ such that, for each $x_2$ in $[c,d]$, the integral $\int_a^b f(t,x_2)dt$ is improper of the second kind and converges pointwies to a function $F$ on $[c,d]$. Assume that $D_2f$ exists and is continuous on $(a,b]\times [c,d]$ and that $\int_a^b D_2 f(t,x_2)dt$ is and improper integral of the second kind which converges uniformly on $[c,d]$ to $F(x_2)$. Then $F$ is differentiable at each $x_2$ in $[c,d]$ and $$F'(x_2)=\int_a^b D_2 f(t,x_2)dt$$

Theorem 13.4.7: Let $f$ be a continuous, real-valued function on $[a,\infty)\times [c,d]$ in $\mathbb{R}^2$. Assume that $\int_a^\infty f(x_1,x_2) dx_1$ is an improper integral of the first kind that converges uniformly for $x_2$ in $[c,d]$ to a function $F(x_2)$. Then $F$ is integrable on $[c,d]$ and $$\int_c^d F(x_2)dx_2 = \int_a^\infty [\int_c^d f(x_1,x_2)dx_2]dx_1$$

Theorem 13.4.8: Let $f$ be a continuous, real-valued function on $(a,b]\times [c,d]$. If the improper integral of the second kind $\int_a^b f(x_1,x_2)dx_1$ converges uniformly to $F(x_2)$ on $[c,d]$, then $F$ is integrable on $[c,d]$ and $$\int_c^d F(x_2)dx_2 = \int_a^b [\int_c^d f(x_1,x_2)dx_2]dx_1$$

Theorem 13.4.9: Let $f$ be continuous on $S=[a,\infty)\times [c,\infty)$. Fot $(x_1,t)$ in $S$, define $g(x_1,t)=\int_c^t f(x_1,x_2)dx_2$. Assume that

i) $F(x_2) = \int_a^\infty f(x_1,x_2)dx_1$ [uniformly] for $x_2$ in $[c,\infty)$.

ii) $G(t)=\int_a^\infty g(x_1,t)dx_1$ [uniformly] for $t$ in $[c,\infty)$.

iii) $H(x_1)=\int_c^\infty f(x_1,x_2)dx_2$ [uniformly] for $x_1$ in $[a,\infty)$.

iv) $\int_a^\infty H(x_1)dx_1 = \int_a^\infty [\int_c^\infty f(x_1,x_2)dx_2] dx_1$ converges.

Then $$\int_a^\infty [\int_c^\infty f(x_1,x_2)dx_2]dx_1 = \int_c^\infty [\int_a^\infty f(x_1,x_2)dx_2]dx_2$$

13.5 Functions Defined by Improper Integrals

Theorem 13.5.1: For all $x,y>0$, $$B(x,y)=\frac{\Gamma (x) \Gamma (y)}{\Gamma (x+y)}$$

13.6 The Laplace Transform

Definition 13.6.1: Let $f$ be a real-valud function on $[0,\infty)$ having the following property: There exists an $s_0 >0$ such that for $s \ge s_0$. the integral $$L f(s) = F(s)=\int_0^\infty e^{-st}f(t)dt$$ converges. Then $L$ is called the Laplace transform, and $Lf=F$ is called the Laplace transform of $f$. The domain of the Laplace transform will be denoted $\mathcal{L}$.

Theorem 13.6.1 The First Shift Theorem: If $f$ is in $\mathcal{L}$, if $F(s)=Lf(s)$, and if $c$ is constant, then $$L(e^{-ct}f(t))(s)=F(s+c)$$

Theorem 13.6.2 The Second Shift Theorem: If $f$ is in $\mathcal{L}$ and if $F=Lf$, then, for any constant $c>0$, $$L(f(t-c))(s)=e^{-cs}F(s)$$

Theorem 13.6.3: If $f$ is in $\mathcal{L}$ and if $F=Lf$, then, for any constant $c>0$, $$L(f(ct))(s)=\frac{1}{c}F(\frac{s}{c})$$

Definition 13.6.2: Let $f$ be a real-valuedfunction defined on $[0,\infty)$ that has the following properties:

i) For any $b>0$, $f$ is bounded on $[0,b]$ and is continuous except possibly at finitely many points in $[0,b]$.

ii) There exists constants $a>0$ and $r$ in $\mathbb{R}$ such that $$\left| f(t)\right| \le ae^{rt}$$ for all $t$ in $[0,\infty)$.

Then $f$ is said to be of exponential type. The collection of all functions of exponential type is denoted $\mathcal{E}$.

Theorem 13.6.4:

i) If $f_1$ and $f_2$ are in $\mathcal{E}$ and if $c_1$ and $c_2$ are in $\mathbb{R}$, then $c_1f_1 + c_2f_2$ and $f_1f_2$ are in $\mathcal{E}$.

ii) If $f$ is in $\mathcal{E}$ and if we define $g(t)=\int_0^t f(u) du$, then $g$ is in $\mathcal{E}$.

Theorem 13.6.5: Suppose that $f$ is continuous and that $f'$ exists at all but countable many discrete points in $[0,\infty)$. If $f'$ is in $\mathcal{E}$, then $f$ is also in $\mathcal{E}$.

Theorem 13.6.6: If $f$ is in $\mathcal{E}$, then the Laplace transform of $f$ exists.

Theorem 13.6.7 Uniqueness Theorem: Suppose that $f$ and $g$ are $\mathcal{E}$ and that $Lf(s)=Lg(s)$ for sufficiently large $s$. Then $f(t)=g(t)$ for all $t$ where $f$ and $g$ are continuous.

Theorem 13.6.8 The First Differentiation Theorem: If $f$ is continuous on $[0,\infty)$ and if $f'$ is in $\mathcal{E}$, then $$(Lf')(s)=s(Lf)(s)-f(0)$$

Corollary 13.6.9: Suppose that $f$ has $k-1$ continuous derivatives on $[0,\infty)$ and that $f^{(k)}$ is in $\mathcal{E}$. Then $$(Lf^{(k)})(s)=s^k(Lf)(s)-\sum_{j=0}^{k-1}s^{k-j-1}f^{(j)}(0)$$

Theorem 13.6.10 The First Integration Theorem: If $f$ is in $\mathcal{E}$, if $F=Lf$, and if $g(t)=\int_0^t f(u)du$, then $$Lg(s)=\frac{F(s)}{s}$$

Theorem 13.6.11 The SecondDifferentiation Theorem: If $f$ is in $\mathcal{E}$ and if $F=Lf$, then $F$ is differentiable and $$F'(s)=-:L(tf(t))(s)$$ for sufficiently large values of $s$.

Theorem 13.6.12 The Second Integration Theorem: If the function $f(t)/t$ is in $\mathcal{E}$ and if $F=Lf$, then for $s$ sufficiently large, $F$ is integrable on $[s,\infty)$ and $$\int_s^\infty F(u)du = L(\frac{f(t)}{t})(s)$$

Theorem 13.6.13: If $f$ is in $\mathcal{E}$ and if $F=Lf$, then there exists an $s_0$ such that, for $s\ge s_0$, $\left sF(s)\right|$ is bounded. In particular, $\lim_{s\rightarrow \infty} F(s)=0$.

Theorem 13.6.14: If $f$ is continuous on $[0,\infty)$, if $f'$ is in $\mathcal{E}$, and if $F=Lf$, then $\lim_{s\rightarrow \infty} sF(s)=f(0)$

Theorem 13.6.15: If $f$ is in $\mathcal{E}$, if $\lim_{t\rightarrow \infty} f(t)=c$, and if $F=Lf$, then $\lim_{s\rightarrow 0^+}sF(s)=c$.

Definition 13.6.3: The convolution of $f$ and $g$, denoted $f\star g$, is the function whose value at $t\ge 0$ is $$(f\star g)(t)  = \int_0^t f(u) g(t-u) du$$

Theorem 13.6.16: Let $f,g,$ and $h$ be functions that are defined on $[0,\infty)$, vanish on $(-\infty,0)$, and, for each $b>0$, are bounded and piecewise continuous on $[0,b]$.

i) The function $f\star g$ is also defined on $[0,\infty)$, vanishes on $(-\infty,0)$, and, for each $b>0$, is continuous and bounded on $[0,b]$.

ii) Convolution is a commutative operation: $f \star g = g \star f$

iii) Convolution is an associative operation: $(f\star g) \star h = f\star (g\star h)$

iv) Convolution distributes over addition:  $f\star (g+h)=f\star g + f\star h$

Theorem 13.6.17: If $f$ and $g$ are in $\mathcal{E}$, then 

i) $f\star g$ also in $\\mathcal{E}$.

ii) $L(f\star g) = (Lf)\cdot (Lg)$

Reference

Douglass - Introduction to Mathematical Analysis