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Chapter 14: Q57P (page 702)

Find the inverse Laplace transform of the following functions by using (7.16). p+1p(p2+1)

Short Answer

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The residues at poles, f(t)=1+sint-cost

Step by step solution

01

To find the poles of    by denominator.

Using convolution of we have to find the inverse transform.

The given equation is,

Rewrite it equation as,

To find the poles of by denominator as,

Simple poles have the above equation, and

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Most popular questions from this chapter

w =√z. Hint: This is equivalent to w2 = z; find x and y in terms of u and v and then solve the pair of equations for u and v in terms of x and y. Note that this is really the same problem as Problem 1 with the z and w planes interchanged.

Evaluate the integrals by contour integration.

I=∫0xcos(θ)dθ5-4cos(θ)

Show that equation (4.4) can be written as (4.5). Then expand each of the fractions in the parenthesis in (4.5) in powers of z and in powers of 1z[see equation (4.7) ] and combine the series to obtain (4.6), (4.8), and (4.2). For each of the following functions find the first few terms of each of the Laurent series about the origin, that is, one series for each annular ring between singular points. Find the residue of each function at the origin. (Warning: To find the residue, you must use the Laurent series which converges near the origin.) Hints: See Problem 2. Use partial fractions as in equations (4.5) and (4.7). Expand a term 1(z-α)in powers of z to get a series convergent for z<α, and in powers of 1z to get a series convergent for z>α.

Using the definition (2.1) of , show that the following familiar formulas hold. Hint : Use the same methods as for functions of a real variable.

27..

In equation (7.18), let u (x) be an even function and υ(x)be an odd function.

  1. If f(x)=u(x)+iυ(x), show that these conditions are equivalent to the equationf*(x)=f(-x) .
  2. Show that

πu(a)=PV∫0∞2xυ(x)x2-a2 dx,      πυ(a)=-PV∫0∞2au(x)x2-a2 dx

These are Kramers-Kroning relations. Hint: To find u(a), write the integral for u(a) in (7.18) as an integral from -∞to 0 plus an integral from 0 to ∞. Then in the to integral -∞to 0, replace x by -x to get an integral from 0 to∞ , and userole="math" localid="1664350095623" υ(-x)=-υ(x) . Add the two to integrals and simplify. Similarly findrole="math" localid="1664350005594" υ(a) .
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