- 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 sinzis 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
- 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 r1 and r2 having potentials U1 and U2, respectively. Find the complex potential. r1 = 0.5 r2 = 2.0, U1 = – 110V, U2 = 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
- 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
- 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
- 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) = iz3. 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 z2? 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
- (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)
- 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
- (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
- (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
- (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 a0 = a1 = 1, an+1 + an + an-1 if Find the limit of the sequence (an+1/an.)(b) Fibonacci’s rabbit problem. Compute a list of a1, ··· a12.
- 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
- 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
- (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
- (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
- 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
- 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
- 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.
- 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.