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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,cyd.
  • 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,r0r,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,r0r,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} +(...resxf(x)dx=re(ss0)x(es0xf(x))dx=0e(ss0)xG(x)dx=e(ss0)xG(x)|r+(ss0)re(ss0)xG(x)dx=e(ss0)rG(r)+(ss0)re(ss0)xG(x)dx,ss0.
  • https://math.libretexts.org/Bookshelves/Analysis/Functions_Defined_by_Improper_Integrals_(Trench)/01%3A_Chapters/1.01%3A_Introduction_to_Improper_Functions
    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+\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)Δybafy(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 yS 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 yS 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}},\...repx|g(x,y)|dx=re(pp0)xep0x|g(x,y)|dxKre(pp0)xdx=Ke(pp0)rpp0, for x sufficiently large if p0>0, Theorem [theorem:4] implies that 0epxxαsinxydx and 0epxxαcosxydx 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^{-...|r1resxf(x)dx|M|e(ss0)r1+e(ss0)r+(ss0)r1re(ss0)xdx|3Me(ss0)r3Me(s1s0)r,ss1. r1resxxnf(x)dx=rn1e(ss0)r1H(r)rne(ss0)rH(r)r1rH(x)(e(ss0)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

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