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Multiplication if the product of two complex numbers is zero, show that at least one factor must be zero.

By definition, the inverse sine w = arc sin *z* is the relation such that sin w = z. The inverse cosine w – arcos z is the relation such that cos w = z. The inverse tangent, increase cotangent, inverse hyperbolic sine, etc, are defined and denoted in a similar fashion. (Note that all these relations are multi valued.) Using sin w = (e^{iw} – e ^{–iw})/(2i) and similar representations of cos, w etc. show that

Between parallel plates at y = x and x + k, potentials 0 and 100 V, respectively

Find the potential between two infinite coaxial cylinders of radii r_{1} and r_{2 }having potentials U_{1} and U_{2}, respectively. Find the complex potential. r_{1} = 0.5 r_{2} = 2.0, U_{1} = – 110V, U_{2 }= 110 V.

Map the upper half z-plane onto the unit disk |w| < 1 so that 0, ∞, – 1 are mapped onto 1, i, –i, respectively. What are the boundary conditions on |w| = 1 resulting from the potential in Prob. 9? What is the potential at w = 0?

Graph the equipotential lines and lines of force in (a) – (d) (four graphs, Re F(z) and 1m F(z) on the same axes). Then explore further complex potentials of your choice with the purpose of discovering configurations that might be of practical interest.

(a) F(z) = z^{2}

(b) F(z) = iz^{2}

(c) F(z) = 1/z

(d) F(z) = i/z

Verify Theorem 1 for Ф* (u, v) = uv, w = f(z) = ez, and D: x < 0, 0 < y < π, Sketch D and D*.

What happens in Prob. 5 if you replace the potential by the conjugate Ф* = 2uv? Sketch or graph some of the equipotential line Ф = const.

Show that in Example 2 the y-axis is mapped onto the unit circle in the w-plane.

Find the temperature and the complex potential in an infinite plate with edges y = x – 2 and y = x + 2 kept at – 10°C and 20°C, respectively.

Interpret Prob. 10 in Sec. 18.2 as a heat flow problem (with boundary temperatures, say, 20oC and 300oC). Along what curves does the heat flow?

(a) A basic building block is shown in Fig. 407. Find the corresponding temperature and complex potential in the upper half-plane.

(b) Conformal mapping, what temperature in the first quadrant of the z-plane is obtained from (a) by the mapping w = a + z2 and what are the transformed boundary conditions?

(c) Superposition, find the temperature T* and the complex potential F* in the upper half-plane satisfying the boundary condition in Fig. 408.

(d) Semi-infinite strip, applying w = cosh z to(c), obtain the solution of the boundary value problem in Fig.409.

Find the temperature T (x, y) in the given thin metal plate whose faces are insulated and whose edges are kept at the indicated temperatures or are insulated as shown.

Sketch or graph the streamlines and equipotential lines of F(z) = iz^{3}. Find V, find all points at which V is parallel to the x-axis.

Show that F(z) = iz 2 models a flow around a corner. Sketch the streamlines and equipotential lines. Find V.

Cylinder what happens in Example 2 if you replace z by z^{2}? Sketch and interpret the resulting flow in the first quadrant.

Aperture show that F(z) = arccosh z gives confocal hyperbolas as stream lines, with foci at z = ±1, and that the flow circulates around an elliptic cylinder or a plate (the segment from – 1 to 1 in Fig. 416).

(a) Mean value property show that the value of a harmonic function Ф at the center of a circle equals the mean of the value of Ф on C (See Sec, 18.4, footnote 1, for definitions of mean values).

(b) Separation of variables, Show that the terms of (7) appear as solutions in separating the Laplace equation in polar coordinates.

(c) Harmonic conjugate, find a series for a harmonic conjugate ψ of Ф from (7).

(d) Power series, find a series for F(z) = Ф + iψ.

Find the equipotential line U = 0V between the cylinders |z| = 0.25 cm and |z| = 4cm kept at – 220 V and 220 V, respectively. (Guess first.)

Find and sketch the equipotential lines of F(z) = (1 + i)/z.

If the region between two concentric cylinders of radii 2 cm and 10 cm contains water and the outer cylinder is kept at 20°C, to what temperature must we heat the inner cylinder in order to have 30°C at distance 5cm from the axis?

(ML-inequality) Find an upper bound of the absolute value of the integral in Prob. 19.

Team project Gain additional insight into the proof of Cauchy’s integral theorem by producing (2) with a contou enclosing z0 (as in Fig. 353) and taking the limit as in the text. Choose and

(c) Two other examples of yourchoice.

(a) Growth of entire functions. If f(z) is not a constant and is analytic for all (finite) z, and R and M are any positive real numbers (no matter how large), show that there exist values of z for which |z| > R and |f(z)| > M.

(b) Growth of polynomials. If f(z) is polynomial of degree n > 0 and M is an arbitrary positive real number (no matter how large), show that there exists a positive real number R such that |f(z)| > M for all |z| > R.

(c) Exponential functions. Show that f(z) = ex has the property characterized in (a) but does not have that characterized in (b).

(d) Fundmental theorem of algebra. If f(z) is a polynomial in z, not a constant, then f(z) = 0 for at least one value of z. Prove this, using (a).

(b) Growth of polynomials. If f(z) is polynomial of degree n > 0 and M is an arbitrary positive real number (no matter how large), show that there exists a positive real number R such that |f(z)| > M for all |z| > R.

(c) Exponential functions. Show that f(z) = ex has the property characterized in (a) but does not have that characterized in (b).

(d) Fundmental theorem of algebra. If f(z) is a polynomial in z, not a constant, then f(z) = 0 for at least one value of z. Prove this, using (a).

(Uniqueness of limit) show that if a sequence converges, its limit is unique

(Multiplication) show that under the assumptions of Prob. 13 the sequence z1z1*, z2z2*, ∙∙∙ converges with the limit ll*.

On Theorem 3 prove that n√n → 1 as n → ∞ (as claimed in the proof of Theorem 3).

(a) The Fibonacci numbers are recursively defined by a_{0} = a_{1} = 1, a_{n+1} + a_{n} + a_{n-1 }if Find the limit of the sequence (a_{n+1}/a_{n}.)

(b) Fibonacci’s rabbit problem. Compute a list of a_{1}, ··· a_{12}. Show that is the number of pairs of rabbits after 12 months if initially there is 1 pair and each pair generates 1 pair per month, beginning in the second month of existence (no deaths occurring).

(c) Generating function. Show that the generating function of the Fibonacci number is f(z) = 1/(1 – z – z^{2}) ; that is, if a power series (1) represents this f(z), its coefficients must be the Fibonacci number and conversely.

Inverse sine developing 1/√1 – z2 and integrating, show that show that this series represents the principal value of arc sin z (defined in Team Project 30, Sec. 13.7)

(c) Generating function. Show that the generating function of the Fibonacci number is f(z) = 1/(1 – z – z2) ; that is, if a power series (1) represents this f(z), its coefficients must be the Fibonacci number and conversely.

Show that |∂un/∂t| < λn2 Ke–λn2to if t > t0 and the series of the expressions on the right converges, by the ratio test, Conclude from this, the Weierstrass test, and Theorem 4 that the series (9) can be differentiated tem by term with respect to t and the resulting series has the sum ∂u/∂t. Show that (9) can be differentiated twice with respect to x and the resulting series has the sum ∂2u/∂x2. Conclude from this and the result to Prob. 19 that (9) is a solution of the heat equation for all t > t0. (The proof that (9) satisfies the given initial condition can be found in Ref. [C10] listed in App.1.)

(a) Uniqueness, prove that the Laurent expansion of a given analytic function in a given annulus is unique.

(b) Accumulation of singularities, does tan (1/z) have a Laurent series that converges in a region 0 > |z| < R? (Give a reason).

(c) Integrals expand the following function in a Laurent series that converges for |z| >0.

(b) Accumulation of singularities, does tan (1/z) have a Laurent series that converges in a region 0 > |z| < R? (Give a reason).

(c) Integrals expand the following function in a Laurent series that converges for |z| >0.

(b) Accumulation of singularities, does tan (1/z) have a Laurent series that converges in a region 0 > |z| < R? (Give a reason).

(c) Integrals expand the following function in a Laurent series that converges for |z| >0.

Graph the orthogonal net of the two families of level curves Re f(z) – const and 1m f(z) = const, where

(a) f(z) = z4,

(b) f(z) = 1/z,

(c) f(z) = 1/z2,

(d) f(z) = (z + i) / (1 + iz). Why do these curves generally intersect at right angles? In your work, experiment to het the best possible graphs. Also do the same for other functions of your own choice. Observe and record shortcomings of your CAS and means to overcome such deficiencies.

(a) f(z) = z4,

(b) f(z) = 1/z,

(c) f(z) = 1/z2,

(d) f(z) = (z + i) / (1 + iz). Why do these curves generally intersect at right angles? In your work, experiment to het the best possible graphs. Also do the same for other functions of your own choice. Observe and record shortcomings of your CAS and means to overcome such deficiencies.

Prove the statement in Prob. 29 for general k = 1, 2, ∙∙∙.

Find a LFT (not w = z) with fixed points 0 and 1.

Find all LFTs whose only fixed point is 0.

Find all LFTS without fixed points in the finite plane.

Inverse find the inverse of the mapping in Example 1. Show that under that inverse the lines x = const are the images of circles in the w-plane with centers on the line v = 1.

Derive the mapping in Example 4 from (2). Find its inverse and prove by calculation that it has the same fixed points as the mapping itself. Is this surprising?

Find all LFTs w(z) that map the x-axis onto u-axis.

Find and sketch or graph the image of the given region under w = sin z. 0 < x < π/6 y arbitrary.

Find and sketch or graph the image of the given region under w = sin z. – π/4 < x < π/4, 0 < y < 3.

Find and sketch or graph the images of the lines x = 0, ± π/6, ± π/3, ± π/2 under the mapping w = sin z.

Find the sketch or graph the image of the given region under w = cos z. 0 < x < 2π, 1/2 < y <1

Find an analytic function w = f(z) that maps;

The interior of the unit circle |z| = 1 onto the exterior of the circle |w + 1| = 5.

The interior of the unit circle |z| = 1 onto the exterior of the circle |w + 1| = 5.

Find an analytic function w = f(z) that maps;

The semi-disk |z| < 1, x > 0 onoto the exterior of the unit circle |w| = 1.

The semi-disk |z| < 1, x > 0 onoto the exterior of the unit circle |w| = 1.

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