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- https://math.libretexts.org/Bookshelves/Analysis/Functions_Defined_by_Improper_Integrals_(Trench)/01%3A_Chapters/1.06%3A_Consequences_of_uniform_convergence\[\begin{aligned} F_{r}(y)&=&F_{r}(y_{0})+\int_{y_{0}}^{y}\left( \int_{a}^{r}f_{y}(x,t)\,dx\right)\,dt\\ &=&F(y_{0})+\int_{y_{0}}^{y}G(t)\,dt \\&&+(F_{r}(y_{0})-F(y_{0})) -\int_{y_{0}}^{y}\left(\int_{...Fr(y)=Fr(y0)+∫yy0(∫rafy(x,t)dx)dt=F(y0)+∫yy0G(t)dt+(Fr(y0)−F(y0))−∫yy0(∫brfy(x,t)dx)dt,c≤y≤d.
- https://math.libretexts.org/Bookshelves/Analysis/Functions_Defined_by_Improper_Integrals_(Trench)/01%3A_Chapters/1.02%3A_Preparation|G(r)−G(r1)|=|(G(r)−L)−(G(r1)−L)|≤|G(r)−L|+|G(r1)−L|<ϵ,r0≤r,r1<b. \[|G(r)|= |G(r_{1})+(G(r)-G(r_{1}))|< |G(r_{1})|+|G(r)-G(r_...|G(r)−G(r1)|=|(G(r)−L)−(G(r1)−L)|≤|G(r)−L|+|G(r1)−L|<ϵ,r0≤r,r1<b. |G(r)|=|G(r1)+(G(r)−G(r1))|<|G(r1)|+|G(r)−G(r1)|≤|G(r1)|+ϵ, |G(r)−G(r1)|=|(G(r)−G(r0))−(G(r1)−G(r0))|≤|G(r)−G(r0)|+|G(r1)−G(r0)|<2ϵ,
- https://math.libretexts.org/Bookshelves/Analysis/Functions_Defined_by_Improper_Integrals_(Trench)/01%3A_Chapters/1.08%3A__Improper_Functions_(Exercises)\[\begin{aligned} \int_{r}^{\infty}e^{-sx}f(x)\,dx&=& \int_{r}^{\infty}e^{-(s-s_{0})x}(e^{-s_{0}x}f(x))\,dx =-\int_{0}^{\infty}e^{-(s-s_{0})x}G'(x)\,dx\\ &=&-e^{-(s-s_{0})x}G(x)\biggr|_{r}^{\infty} +(...∫∞re−sxf(x)dx=∫∞re−(s−s0)x(e−s0xf(x))dx=−∫∞0e−(s−s0)xG′(x)dx=−e−(s−s0)xG(x)|∞r+(s−s0)∫∞re−(s−s0)xG(x)dx=e−(s−s0)rG(r)+(s−s0)∫∞re−(s−s0)xG(x)dx,s≥s0.
- https://math.libretexts.org/Bookshelves/Analysis/Functions_Defined_by_Improper_Integrals_(Trench)/01%3A_Chapters/1.01%3A_Introduction_to_Improper_Functions83), if x∈[a,b] and y, y+Δy∈[c,d], there is a y(x) between y and y+Δy such that \[f(x,y+\Delta y)-f(x,y)=f_{y}(x,y)\Delta y= f_{y}(x,y(x))\Delta y+(f_{y}(x,y(x)...83), if x∈[a,b] and y, y+Δy∈[c,d], there is a y(x) between y and y+Δy such that f(x,y+Δy)−f(x,y)=fy(x,y)Δy=fy(x,y(x))Δy+(fy(x,y(x)−fy(x,y))Δy. |F(y+Δy)−F(y)Δy−∫bafy(x,y)dx|≤∫ba|fy(x,y(x))−fy(x,y)|dx.
- https://math.libretexts.org/Bookshelves/Analysis/Functions_Defined_by_Improper_Integrals_(Trench)/01%3A_Chapters/1.03%3A_Uniform_convergence_of_improper_integrals[definition:3] Let f=f(x,y) be defined on (a,b)×S, where −∞≤a<b≤∞. Suppose f is locally integrable on (a,b) for all y∈S and let c be an arbitrary ...[definition:3] Let f=f(x,y) be defined on (a,b)×S, where −∞≤a<b≤∞. Suppose f is locally integrable on (a,b) for all y∈S and let c be an arbitrary point in (a,b). Then ∫baf(x,y)dx is said to converge uniformly on S if ∫caf(x,y)dx and ∫bcf(x,y)dx both converge uniformly on S.
- https://math.libretexts.org/Bookshelves/Analysis/Functions_Defined_by_Improper_Integrals_(Trench)This is a supplement to the author’s Introduction to Real Analysis text.
- https://math.libretexts.org/Bookshelves/Analysis/Functions_Defined_by_Improper_Integrals_(Trench)/01%3A_Chapters/1.04%3A_Absolutely_Uniformly_Convergent_Improper_Integrals\[\begin{aligned} \int_{r}^{\infty}e^{-px} |g(x,y)|\,dx &=& \int_{r}^{\infty} e^{-(p-p_{0})x}e^{-p_{0}x}|g(x,y)|\,dx\\ &\le& K\int_{r}^{\infty} e^{-(p-p_{0})x}\,dx= \frac{K e^{-(p-p_{0})r}}{p-p_{0}},\...∫∞re−px|g(x,y)|dx=∫∞re−(p−p0)xe−p0x|g(x,y)|dx≤K∫∞re−(p−p0)xdx=Ke−(p−p0)rp−p0, for x sufficiently large if p0>0, Theorem [theorem:4] implies that ∫∞0e−pxxαsinxydx and ∫∞0e−pxxαcosxydx converge absolutely uniformly on (−∞,∞) if p>0 and α ≥ 0.
- https://math.libretexts.org/Bookshelves/Analysis/Functions_Defined_by_Improper_Integrals_(Trench)/01%3A_Chapters/1.07%3A_Applications_to_Laplace_transforms\[\begin{aligned} \left|\int_{r}^{r_{1}}e^{-sx}f(x)\,dx\right|&\le& M\left|e^{-(s-s_{0})r_{1}} +e^{-(s-s_{0})r} +(s-s_{0})\int_{r}^{r_{1}}e^{-(s-s_{0})x}\,dx\right|\\ &\le &3Me^{-(s-s_{0})r}\le 3Me^{-...|∫r1re−sxf(x)dx|≤M|e−(s−s0)r1+e−(s−s0)r+(s−s0)∫r1re−(s−s0)xdx|≤3Me−(s−s0)r≤3Me−(s1−s0)r,s≥s1. ∫r1re−sxxnf(x)dx=rn1e−(s−s0)r1H(r)−rne−(s−s0)rH(r)−∫r1rH(x)(e−(s−s0)xxn)′dx,
- https://math.libretexts.org/Bookshelves/Analysis/Functions_Defined_by_Improper_Integrals_(Trench)/01%3A_Chapters/1.05%3A_Dirichlet%E2%80%99s_Tests\[\begin{aligned} \int_{r}^{r_{1}}g(x,y)h(x,y)\,dx&=&\int_{r}^{r_{1}}g(x,y)H_{x}(x,y)\,dx \nonumber\\ &=&g(r_{1},y)H(r_{1},y)-g(r,y)H(r,y)\label{eq:21}\\ &&-\int_{r}^{r_{1}}g_{x}(x,y)H(x,y)\,dx. \nonu...∫r1rg(x,y)h(x,y)dx=∫r1rg(x,y)Hx(x,y)dx=g(r1,y)H(r1,y)−g(r,y)H(r,y)−∫r1rgx(x,y)H(x,y)dx. ∫r1rcosxyx+ydx=sinxyy(x+y)|r1r+∫r1rsinxyy(x+y)2dx=sinr1yy(r1+y)−sinryy(r+y)+∫r1rsinxyy(x+y)2dx.
- https://math.libretexts.org/Bookshelves/Analysis/Functions_Defined_by_Improper_Integrals_(Trench)/01%3A_Chapters