### Harmonic Analysis, Convolution and L^1

Let be a positive function in . Define a new function by Prove that . Please see the attached file for the fully formatted problems.

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Let be a positive function in . Define a new function by Prove that . Please see the attached file for the fully formatted problems.

Find the integral of: ((1+(sinh t)^2)^(1/2))dt.

Let X be a normed space, I closed interval ( or half-open on the right) and a = inf I, b = sup I. Let h : I -> [0,infinity) be a continuous function such that integral ( from a to b ) h(t)dt < positive infinity where integral from a to b represents the improper integral when I is not closed. Let epsilon > 0 and

If is a measure space and , show that defines a bounded integral operator. Please see the attached file for the fully formatted problems.

Please see the attached file for the fully formatted problems.

Find the volume of the solid formed when the region described is revolved about the x axis using washers and disks. 14) the region under the curve y= cubed root of x on the interval 0≤x≤8. 16) the region bounded by the lines x=0, x=1, y=x+1, and y=x+2. 20) the region bounded by the curves y=e^x and y=e^-x on

Which is the acceptable trial solution? (See attached file for full problem description)

Show that the solution of: dM/dt = Poert - pM M(0)=0 is M(t) = [Po/(r + p)](ert - e-pt) Please see the attached file for the fully formatted problems.

1. Use integration to find a general solution of the differential equation. dy / dx = (x-2)/ x = 1 - 2/x 2.Solve the differential equation. dy / dx = x + 2

Let H be the collection of all absolutely continuous functions f [0,1] -> F, where F denotes either real or complex field ) such that f (0) = 0 and . If for f andg in H, then H is a Hilbert space. Please see the attached file for the fully formatted problem.

Please see attached

Hello. Thank you for taking the time to help me. I cannot use mathematical symbols, thus, * will denote a partial derivative. For example, u*x denotes the partial derivative of u with respect to x. To simplify things, I will let p=u*x and q=u*y. Furthermore, I will use ^ to denote a power. For example, x^2 means x squared. Also,

Please see attached.

Hello. Thank you for taking the time for helping me. The following is the problem which I need to solve (there are actually two parts): I need to construct Green's function for the Dirichlet problem (Laplace's equation) in the upper half plane R={(x,y) : y>0} and I must derive Poisson's integral formula for the half plane.

--- Using the method of residues, verify the following: --- (See attached file for full problem description)

Prove the following: Let be a sequence of functions continuous on a set containing the contour , and suppose that converges uniformly to on . Then the series converges to . Using this result and the Generalized Cauchy Integral formula for derivatives (see below), show the following: If all are analytic

Please see the attached file for the fully formatted problems.

Please see the attached file for the fully formatted problems.

Use residues to evaluate integral of a trigonometric function. Please see the attached file for the fully formatted problems.

Please see the attached file for the fully formatted problems.

Let g be continuous on the real interval [0,1] and define H(z) := integral (from 0 to 1) [g(t)/(1-z[t^2])]dt, (|z| < 1) Prove that H is analytic in the open disk |z| < 1.

Find the area of the region bounded between two given curves by integration. {(x,y): Y^2 ≤ 4x, 4x^2 + 4y^2 ≤9}

A) (e^x/((e^x+2)^(1/2))dx between 0.5,0 b) (x^2/((1-2x^3)^(1/2))dx c) (sin^5(x))dx between ((3.14/2),0) d) (x^2 cos(x))dx e) (1/((2x+5)(1-3x)))dx f) (3x-4/((x-2)(x+1))) dx between the limits 5,4 g) (4x.e^(-4x))dx between the limits 1,0 h) (3x.sin(3x))dx i) (sin(4x) - 4cos(3x)

Please see attached. Hi, I am having trouble doing these problems listed below. Please show me how to solve these problems for future reference. Thank you very much. I would like for you to show me all of your work/calculations and the correct answer to each problem. For Exercise 2, find the mode of the probabili

I am looking for the solution of this transformation I need a detailed solution. Also I would like to see the original formula for the Laplace transformations needed. If f(t) is a periodic, continuous function with period T>0, show that its Laplace transform is... Please see attached.

Without evaluating the integral show that (see attachment) when C is the same arc as the one in Example 1 (see attachment for example) Please see the attached file for the fully formatted problems.

1. Suppose f: [a,b]  is a function such that f(x)=0 for every x (a,b]. a) Let  > 0. Choose n   such that a + 1/n < b and |f(a)|/n <. Let P ={a, a+1/n, b}  ([a,b]). Compute (f,P) - (f,P) and show that is less than . b) Prove

(1) Find the volume of the solid bounded by the paraboloid x2 + y2 = 2z, the plane z = 0 and the cylinder x2 + y2 = 9. (2) Find the volume of the region in the first octant bounded by x + 2y + 3z = 6. (3) Find the area of the solid that is bounded by the cylinders x2+z2 = r2 and y2+z2 = r2. (4) Find the volume enclosed by t

Showing that a particular differential form is closed but not exact, as an application of Stokes' theorem (differential forms version)

Please see the attached file for the fully formatted problems.