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calculus

Questions and Answers of Calculus

Let u and v denote the real and imaginary components of the function f defined by means of the equationsVerify that the Cauchy-Riemann equations ux = vy and uy = ˆ’vx are satisfied at the
Solve equations (2), Sec. 23 for ux and uy to show thatThen use these equations and similar ones for vx and vy to show that in Sec. 23 equations (4) are satisfied at a point z0 if equations (6) are
Let a function f(z) = u + iv be differentiable at a nonzero point z0 = r0 exp(iθ0). Use the expressions for ux and vx found in Exercise 7, together with the polar form (6), Sec. 23, of the
Apply the theorem in Sec. 22 to verify that each of these functions is entire: (a) f (z) = 3x + y + i(3y − x); (b) f (z) = sin x cosh y + i cos x sinh y; (c) f (z) = e−y sin x − ie−y cos
With the aid of the theorem in Sec. 21, show that each of these functions is nowhere analytic: (a) f (z) = xy + iy; (b) f (z) = 2xy + i(x2 − y2);
Let a function f be analytic everywhere in a domain D. Prove that if f (z) is realvalued for all z in D, then f (z) must be constant throughtout D.
Show that u(x, y) is harmonic in some domain and find a harmonic conjugate v(x, y) when (a) u(x, y) = 2x(1 − y); (b) u(x, y) = 2x − x3 + 3xy2; (c) u(x, y) = sinh x sin y; (d) u(x, y) = y/(x2 +
Show that if v and V are harmonic conjugates of u(x, y) in a domain D, then v(x, y) and V (x, y) can differ at most by an additive constant.
Suppose that v is a harmonic conjugate of u in a domain D and also that u is a harmonic conjugate of v in D. Show how it follows that both u(x, y) and v(x, y) must be constant throughout D.
Let the function f (z) = u(r, θ) + iv(r, θ) be analytic in a domain D that does not include the origin. Using the Cauchy-Riemann equations in polar coordinates (Sec. 23) and assuming continuity of
Verify that the function u(r, θ) = ln r is harmonic in the domain r > 0, 0 < θ < 2π by showing that it satisfies the polar form of Laplace's equation, obtained in Exercise 5. Then use the
Show that (a) exp (2 ± 3πi) = -e2; (b) exp (2 + πi/4) = √e/2 (1 + i); (c) exp(z + πi) = - exp z.
(a) Show that if ez is real, then Im z = nπ (n = 0, ±1, ±2, . . .). (b) If ez is pure imaginary, what restriction is placed on z?
Write Re(e1/z) in terms of x and y. Why is this function harmonic in every domain that does not contain the origin?
Let the function f (z) = u(x, y) + iv(x, y) be analytic in some domain D. State why the functions U(x, y) = eu(x,y) cos v(x, y), V (x, y) = eu(x,y) sin v(x, y) are harmonic in D and why V (x, y) is,
Establish the identity (ez)n = enz (n = 0, ±1, ±2, . . .) in the following way. (a) Use mathematical induction to show that it is valid when n = 0, 1, 2, . . . . (b) Verify it for negative integers
Use the Cauchy-Riemann equations and the theorem in Sec. 21 to show that the function f (z) = exp z is not analytic anywhere.
Show in two ways that the function f(z) = exp(z2) is entire. What is its derivative?
Write |exp(2z + i)| and |exp(iz2)| in terms of x and y. Then show that |exp(2z + i) + exp(iz2)| ≤ e2x + e−2xy.
Show that |exp(z2)| ≤ exp(|z|2).
Prove that |exp(− 2z)| < 1 if and only if Re z > 0.
Find all values of z such that (a) ez = −2; (b) ez = 1 +√3i; (c) exp(2z − 1) = 1.
Show that exp(iz) = exp(i) if and only if z = nπ (n = 0, ±1, ±2, . . .).
Show that (a) Log(-ei) = 1 - π/2i; (b) Log (1 - i) = ½ ln 2 - π/4 i.
Show in two ways that the function ln(x2 + y2) is harmonic in every domain that does not contain the origin.
Show that (a) log e = 1 + 2nπi (n = 0,±1,±2, . . .); (b) log I = (2n + ½) πi (n = 0, ±1, ±2, ...); (c) log (- 1 + √3i)} = ln 2 + 2(n + 1/3) πi (n = 0, ±1, ±2, ...).
Show that (a) Log(1 + i)2 = 2 Log(1 + i); (b) Log(−1 + i)2 ≠ 2 Log(−1 + i).
Show that (a) log(i2) = 2 log i when log z = ln r + iθ (r > 0, π/4 < θ < 9π/4); (b) log(i2) ≠ 2 log i when log z = ln r + iθ (r > 0, 3π/4 < θ < 11π/4).
Show that (a) The set of values of log(i1/2) is (n + 1/4)πi (n = 0, ±1, ±2, ...) and that the same is true of (1/2) log i ; (b) The set of values of log(i2) is not the same as the set of values of
Show that if Re z1 > 0 and Re z2 > 0, then Log(z1z2) = Log z1 + Log z2.
Verify expression (4), Sec. 32, for log(z1/z2) by (a) Using the fact that arg(z1/z2) = arg z1 − arg z2 (Sec. 8); (b) Showing that log(1/z) = −logz (z ≠ 0), in the sense that log(1/z) and −log
Show that property (6), Sec. 32, also holds when n is a negative integer. Do this by writing z1/n = (z1/m)−1 (m = −n), where n has any one of the negative values n = −1,−2, . . . (see
Show that (a) (1 + i)i = exp (-π/4 + 2nπ) exp (I ln 2/2) (n = 0, ±1, ±2, ...); (b) (−1)1/π = e(2n+1)i (n = 0,±1,±2, . . .).
Find the principal value of (a) ii; (b) [e/2(- 1 - √3i)3πi; (c) (1 - i)4i.
Use definition (1), Sec. 33, of zc to show that (−1 + √3i)3/2 = ± 2 √2. Use definition (1) zc = ec log z, where log z denotes the multiple-valued logarithmic function. Equation (1) provides a
Show that the principal nth root of a nonzero complex number z0 that was defined in Sec. 9 is the same as the principal value of z01/n defined by equation (5), Sec. 33. P.V. zc = ec Log z. Equation
Let c = a + bi be a fixed complex number, where c ≠ 0, ±1, ±2, . . . , and note that ic is multiple-valued. What additional restriction must be placed on the constant c so that the values of |ic|
Give details in the derivation of expressions (2), Sec. 34, for the derivatives of sin z and cos z. d/dz sin z = cos z and d/dz cos z = - sin z. It is easy to see from definitions (1) that the sine
Show that (a) cos(iz) = cos(iz) for all z; (b) sin(iz) = sin(iz) if and only if z = nπi (n = 0, ±1, ±2, . . .).
Find all roots of the equation sin z = cosh 4 by equating the real parts and then the imaginary parts of sin z and cosh 4.
According to the final result in Exercise 2(b), sin(z + z2) = sin z cos z2 + cos z sin z2. By differentiating each side here with respect to z and then setting z = z1, derive the expression cos(z1 +
Use identity (9) in Sec. 34 to show that (a) 1 + tan2 z = sec2 z; (b) 1 + cot2 z = csc2 z.
In Sec. 34, use expressions (13) and (14) to derive expressions (15) and (16) for |sin z|2 and |cos z|2. (13) sin z = sin x cosh y + i cos x sinh y, (14) cos z = cos x cosh y − i sin x sinh
Point out how it follows from expressions (15) and (16) in Sec. 34 for |sin z|2 and |cos z|2 that (a) |sin z| ≥ |sin x|; (b) |cos z| ≥ |cos x|.
With the aid of expressions (15) and (16) in Sec. 34 for |sin z|2 and |cos z|2, show that (a) |sinh y| ≤ |sin z| ≤ cosh y; (b) |sinh y| ≤ |cos z| ≤ cosh y.
Verify that the derivatives of sinh z and cosh z are as stated in equations (2), Sec. 35.Equation (2)Because of the way in which the exponential function appears in definitions (1) and in the
Show how identities (6) and (8) in Sec. 35 follow from identities (9) and (6), respectively, in Sec. 34. (6) cosh2 z − sinh2 z = 1, (8) cosh(z1 + z2) = cosh z1 cosh z2 + sinh z1 sinh z2 (6) cosh2 z
Show that |sinh x| ≤ |cosh z| ≤ cosh x by using (a) Identity (12), Sec. 35; (12) |cosh z|2 = sinh2 x + cos2 y, where z = x + iy. While these identities follow directly from definitions (1), they
Show that (a) sinh(z + πi) = −sinh z; (b) cosh(z + πi) = cosh z; (c) tanh(z + πi) = tanh z.
Using the results proved in Exercise 8, locate all zeros and singularities of the hyperbolic tangent function. (14) sinh z = 0 if and only if z = nπi (n = 0, ±1, ±2, . . .) (15) cosh z = 0 if and
Use rules in calculus to establish the following rules when w(t) = u(t) + iv(t) is a complex-valued function of a real variable t and w'(t) exists: (a) d/dt w(-t) = - w'(-t) where w'(-t) denotes the
Show that if m and n are integers,
According to definition (2), Sec. 38, of definite integrals of complex-valued functions of a real variable,Evaluate the two integrals on the right here by evaluating the single integral on the left
Let w(t) = u(t) + iv(t) denote a continuous complex-valued function defined on an interval ˆ’a ‰¤ t ‰¤ a.(a) Suppose that w(t) is even; that is, w(ˆ’t) =
Show that if w(t) = u(t) + iv(t) is continuous on an interval a ‰¤ t ‰¤ b, then(a)(b) where Ï†(Ï„) is the function in equation (9), Sec. 39.
Derive the equation of the line through the points (Î±, a) and (Î², b) in the Ï„t plane that are shown in Fig. 37. Then use it to find the linear function
Verify expression (14), Sec. 39, for the derivative of Z(Ï„) = z[Ï†(Ï„)].Z'(Ï„ ) = z '[Ï†(Ï„)]Ï†'(Ï„ )enables us to
Suppose that a function f (z) is analytic at a point z0 = z(t0) lying on a smooth arc z = z(t) (a ≤ t ≤ b). Show that if w(t) = f [z(t)], then w'(t) = f'[z(t)]z'(t) when t = t0. Suggestion: Write
(z) = (z + 2)/z and C is (a) The semicircle z = 2 eiθ (0 ≤ θ ≤ π); (b) The semicircle z = 2 eiθ (π ≤ θ ≤ 2π); (c) The circle z = 2 eiθ (0 ≤ θ ≤ 2π).
Let C0 and C denote the circlesz = z0 + ReiÎ¸ (ˆ’Ï€ ‰¤ Î¸ ‰¤ Ï€) and z = ReiÎ¸ (ˆ’Ï€
(a) Suppose that a function f (z) is continuous on a smooth arc C, which has a parametric representation z = z(t) (a ‰¤ t ‰¤ b); that is, f [z(t)] is continuous on the
f(z) = z − 1 and C is the arc from z = 0 to z = 2 consisting of (a) The semicircle z = 1 + eiθ (π ≤ θ ≤ 2π); (b) The segment z = x (0 ≤ x ≤ 2) of the real axis.
f (z) = π exp(π) and C is the boundary of the square with vertices at the points 0, 1, 1 + i, and i, the orientation of C being in the counterclockwise direction.
f (z) is defined by means of the equationsand C is the arc from z = ˆ’1 ˆ’ i to z = 1 + i along the curve y = x3.
f (z) = 1 and C is an arbitrary contour from any fixed point z1 to any fixed point z2 in the z plane.
f (z) is the branch z−1+i = exp[(−1 + i)log z] (|z| > 0, 0 < argz < 2π) of the indicated power function, and C is the unit circle z = eiθ (0 ≤ θ ≤ 2π).
With the aid of the result in Exercise 3, Sec. 38, evaluate the integralwhere m and n are integers and C is the unit circle |z| = 1, taken counterclockwise.
Evaluate the integral I in Example 1, Sec. 41, using this representation for C: z = √4 − y2 + iy (−2 ≤ y ≤ 2). (See Exercise 2, Sec. 39.)
Without evaluating the integral, show thatwhen C is the same arc as the one in Example 1, Sec. 43.
Let C denote the line segment from z = i to z = 1. By observing that of all the points on that line segment, the midpoint is the closest to the origin, show thatwithout evaluating the integral.
Show that if C is the boundary of the triangle with vertices at the points 0, 3i, and ˆ’4, oriented in the counterclockwise direction (see Fig. 48), then
Let CR denote the upper half of the circle |z| = R (R > 2), taken in the counterclockwise direction. Show thatThen, by dividing the numerator and denominator on the right here by R4, show that the
Let CR be the circle |z| = R (R > 1), described in the counterclockwise direction. Show thatand then use l'Hospital's rule to show that the value of this integral tends to zero as R tends to infinity.
Apply inequality (1), Sec. 43, to show that for all values of x in the interval ˆ’1 ‰¤ x ‰¤ 1, the functionsˆ—satisfy the inequality |Pn(x)| ‰¤
Use an antiderivative to show that for every contour C extending from a point z1 to a point z2,
By finding an antiderivative, evaluate each of these integrals, where the path is any contour between the indicated limits of integration:(a)(b) (c)
Use the theorem in Sec. 44 to show thatwhen C0 is any closed contour which does not pass through the point z0. [Compare with Exercise 10(b), Sec. 42.]
Show that ∫1-1 zi dz = 1 + e-π/2 (1 - i), where the integrand denotes the principal branch zi = exp(i Log z) (|z| > 0,−π < Arg z < π) of zi and where the path of integration is any contour
Use definition (2), Sec. 55, of limits of sequences to verify the limit of the sequence zn (n = 1, 2( ( ( () found in Example 2, Sec. 55.
Let (n (n = 1, 2( ( ( () denote the principal arguments of the numbersPoint out why And compare with Example 2, Sec. 55.
Use the inequality (see Sec. 4) ||zn| ˆ’ |z|| ‰¤ |zn ˆ’ z| to show that if
Write z = rei(, where 0when 0
Show that if
By recalling the corresponding result for series of real numbers and referring to the theorem in Sec. 56, show that if
Obtain the Maclaurin series representation
Show that when 0
Obtain the Taylor series
Find the Maclaurin series expansion of the function
Use representation (2), Sec. 59, for sin z to write the Maclaurin series for the function((z) = sin(z2),And point out how it follows that
Derive the Taylor series representationStart by writing
Use the identity sinh(z + πi) = −sinh z, verified in Exercise 7(a), Sec. 35, and the fact that sinh z is periodic with period 2πi to find the Taylor series for sinh z about the point z0 = πi.
Find the Laurent series that represents the functionin the domain 0
(a) Let z be any complex number, and let C denote the unit circlein the w plane. Then use that contour in expression (5), Sec. 60, for the coefficients in a Laurent series, adapted to such series
(a) Let f (z) denote a function which is analytic in some annular domain about the origin that includes the unit circle z = ei( (ˆ’Ï€ ‰¤ ( ‰¤
Find a representation for the functionin negative powers of z that is valid when 1
Give two Laurent series expansions in powers of z for the functionAnd specify the regions in which those expansions are valid.
Represent the function(a) By its Maclaurin series, and state where the representation is valid; (b) By its Laurent series in the domain 1
Write the two Laurent series in powers of z that represent the functionIn certain domains, and specify those domains.
(a) Let a denote a real number, where ˆ’1(b) After writing z = eiÎ¸ in the equation obtained in part (a), equate real parts and then imaginary parts on each side of the result
By differentiating the Maclaurin series representationObtain the expansions And

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