Evaluate tan(z) dz with the circle

In each of Problems 1 through 12, determine all singularities of the function and classify each sicos(z)/z2

In each of Problems 1 through 12, determine all singularities of the function and classify each si4 sin(z + 2) (z + i)2(z i)

In each of Problems 1 through 12, determine all singularities of the function and classify each sie1/z (z + 2i)

In each of Problems 1 through 12, determine all singularities of the function and classify each sisin(z)/(z )

In each of Problems 1 through 12, determine all singularities of the function and classify each sicos(2z) (z 1)2(z2 + 1)

In each of Problems 1 through 12, determine all singularities of the function and classify each si/(z + 1)2

In each of Problems 1 through 12, determine all singularities of the function and classify each siz i)/(z2 + 1)

In each of Problems 1 through 12, determine all singularities of the function and classify each sisin(z)/ sinh(z)

In each of Problems 1 through 12, determine all singularities of the function and classify each siz/(z4 1)

In each of Problems 1 through 12, determine all singularities of the function and classify each sitan(z)

In each of Problems 1 through 12, determine all singularities of the function and classify each sisec(z)

In each of Problems 1 through 12, determine all singularities of the function and classify each sie1/z(z+1)

Let f be differentiable at z0 and f (z0) =0. Let g have a pole of order m at z0. Show that f g has a pole of order m at z0.

In each of Problems 1 through 16, use the residue theorem to evaluate the integral 1 + z2 (z 1)2(z + 2i) dz with the circle of radius 7 about i

In each of Problems 1 through 16, use the residue theorem to evaluate the integral 2z (z i)2 dz with the circle of radius 3 about 1.

In each of Problems 1 through 16, use the residue theorem to evaluate the integral ez z dz with the circle of radius 2 about 3i

In each of Problems 1 through 16, use the residue theorem to evaluate the integral cos(z) 4 + z2 dz with the square of side length 3 and sides parallel to the axes centered at 2i.

In each of Problems 1 through 16, use the residue theorem to evaluate the integral z + i z2 + 6 dz with the square of side length 8 and sides parallel to the axes centered at the origin.

In each of Problems 1 through 16, use the residue theorem to evaluate the integral.  z i 2z + 1 dz with the circle of radius 1 about the origin.

In each of Problems 1 through 16, use the residue theorem to evaluate the integral z sinh2 (z) dz with the circle of radius 1 about 1/2.

In each of Problems 1 through 16, use the residue theorem to evaluate the integral cos(z) zez dz with the circle of radius 1/2 about i/8.

In each of Problems 1 through 16, use the residue theorem to evaluate the integral i z (z2 + 9)(z i) dz with the circle of radius 2 about 3

In each of Problems 1 through 16, use the residue theorem to evaluate the integral e2/z2 dz with the square with sides parallel to the axes and of length 3 centered at i.

In each of Problems 1 through 16, use the residue theorem to evaluate the integral 8z 4i + 1 z + 4i dz with the circle of radius 2 about i

In each of Problems 1 through 16, use the residue theorem to evaluate the integral z2 z 1 + 2i dz with the square of side length 4 and sides parallel to the axes centered at 1 2

In each of Problems 1 through 16, use the residue theorem to evaluate the integral coth(z) dz with the circle of radius 2 about i.

In each of Problems 1 through 16, use the residue theorem to evaluate the integral.  (1 z)2 z3 8 dz with the circle of radius 2 about 2

In each of Problems 1 through 16, use the residue theorem to evaluate the integral e2z z(z 4i) dz with any closed path enclosing 0 and 4i.

In each of Problems 1 through 16, use the residue theorem to evaluate the integral z z 1 2 dz with any closed path enclosing 1

Let h and g be differentiable at z0 and g(z0) = 0. Suppose h has a zero of order 2 at z0. Show that Res(g(z)/h(z),z0) = 2g (z0) h(z0) 2 3 g(z0)h(3) (z0) (h(z0))2 . Hint: Use Theorem 22.6. Begin by writing h(z) = (z z0)2 (z) where (z0) = 0.

Suppose f is differentiable at points on a closed path and at all points in the region G enclosed by , except possibly at a finite number of poles of f in G. Let Z be the number of zeros of f in G and P be the number of poles of f in G with each zero and pole counted as many times as its multiplicity. Show that 1 2i  f  (z) f (z) dz = Z P. This formula is known as the argument principle. Hint: If f has a zero of order k at z0, show by looking at the Taylor expansion of f (z) about z0 that f  (z) f (z) = k z z0 + g (z) g(z) where g is differentiable at z0 and g(z0) = 0. Use this to evaluate Res( f  / f,z0). If f has a pole of order m at z1, show by examining the Laurent expansion of f (z) about z1 that f  (z) f (z) = m z z1 + h (z) h(z) for some h(z) that is differentiable and nonzero at z1. Use these facts and the residue theorem to derive the argument principle.

Evaluate  z 2 + z2 dz with as the circle |z| = 2 first by using the residue theorem and then by using the argument principle.

Evaluate  tan(z) dz with the circle |z| = first by using the residue theorem and then by using the argument principle.

Evaluate  z + 1 z2 + 2z + 4 dz with the circle |z| = 2, first by using the residue theorem and then by using the argument principle.

Let p(z) = (z z1)(z z2)(z zn ) with z1, ,zn distinct complex numbers. Let be a positively oriented closed path enclosing each zj . Evaluate  p (z) p(z) dz first by using the residue theorem and then by using the argument principle.

In each of Problems 1 through 10, evaluate the integral. Wherever they appear, and are positive numbers.2 0 1 2 cos( ) d

In each of Problems 1 through 10, evaluate the integral. Wherever they appear, and are positive numbers. 1 x 4 + 1 dx

In each of Problems 1 through 10, evaluate the integral. Wherever they appear, and are positive numbers. 1 x 6 + 1 dx

In each of Problems 1 through 10, evaluate the integral. Wherever they appear, and are positive numbers.2 0 1 6 + sin( ) d

In each of Problems 1 through 10, evaluate the integral. Wherever they appear, and are positive numbers. x sin(2x) x 4 + 16 dx

In each of Problems 1 through 10, evaluate the integral. Wherever they appear, and are positive numbers. 1 x 2 2x + 6 dx

In each of Problems 1 through 10, evaluate the integral. Wherever they appear, and are positive numbers. cos2 (x) (x 2 + 4)2 dx

In each of Problems 1 through 10, evaluate the integral. Wherever they appear, and are positive numbers.2 0 2 sin( ) 2 + sin2 ( ) d

In each of Problems 1 through 10, evaluate the integral. Wherever they appear, and are positive numbers. x 2 (x 2 + 4)2 dx

In each of Problems 1 through 10, evaluate the integral. Wherever they appear, and are positive numbers. cos(x) (x 2 + 2)2 dx

Show that cos(x) x 2 + 1 dx = e.

Show that cos(x) x 2 + 1 dx = e.

Let =. Show that 2 0 1 2 cos2( ) + 2 sin2 ( ) d = 2

Show that /2 0 1 + sin2 ( ) d = 2 (1 + ).

Show that 0 ex2 cos(2x) dx = 2 e2 . Hint: Integrate ez2 about the rectangular path having corners at R and R + i. Use Cauchys theorem to evaluate this integral, set this equal to the sum of the integrals on the sides of the rectangle, and take the limit as R . Assume the standard result that 0 ex2 dx = 2 .

Derive Fresnels integrals: 0 cos(x 2 ) dx = 0 sin(x 2 ) dx = 1 2  2 . Hint: Integrate ei z2 over the closed path bounding the sector 0 x R, 0 /4, as in Figure 22.3. Use Cauchys theorem to evaluate this integral, then evaluate it as the sum of the integrals over the boundary segments of the sector. Show that the integral over the circular arc tends to zero as R , and use the integrals over the line segments to obtain Fresnels integrals. x y R /4

Let and be positive numbers. Show that 0 x sin(x) x 4 + 4 dx = 22 e/2 sin 2  .

Let 0 <<. Show that 0 1 ( + cos( ))2 d = (2 2)3/2

In each of Problems 1 through 10, use Theorem 22.7 to find the inverse Laplace transform of the function.z z2 + 9

In each of Problems 1 through 10, use Theorem 22.7 to find the inverse Laplace transform of the function.1 (z + 3)2

In each of Problems 1 through 10, use Theorem 22.7 to find the inverse Laplace transform of the function.1 (z 2)2(z + 4)

In each of Problems 1 through 10, use Theorem 22.7 to find the inverse Laplace transform of the function.1 (z2 + 9)(z 2)2

In each of Problems 1 through 10, use Theorem 22.7 to find the inverse Laplace transform of the function.1 (z + 5)3

In each of Problems 1 through 10, use Theorem 22.7 to find the inverse Laplace transform of the function.z3 + 8

In each of Problems 1 through 10, use Theorem 22.7 to find the inverse Laplace transform of the function.1 z4 + 1

In each of Problems 1 through 10, use Theorem 22.7 to find the inverse Laplace transform of the function.1 ez(z 1)

In each of Problems 1 through 10, use Theorem 22.7 to find the inverse Laplace transform of the function.z2 (z 2)3

In each of Problems 1 through 10, use Theorem 22.7 to find the inverse Laplace transform of the function.z + 3 (z3 1)(z + 2)