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    MTL 506 - Tutorial Sheet 3

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    vamgadu
    This document provides a tutorial sheet for the course MTL 506: Complex Analysis. It includes 21 problems covering topics such as: proving properties of analytic functions; finding radii of convergence of power series; properties of functions like the logarithm and tangent; mapping regions via Möbius transformations; and properties that characterize different types of Möbius transformations. The problems progress from simpler exercises to more complex proofs and geometric interpretations involving concepts like branches of the logarithm and conformal mappings.

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    MTL 506 - Tutorial Sheet 3

    Uploaded by

    vamgadu
    47 views2 pages
    This document provides a tutorial sheet for the course MTL 506: Complex Analysis. It includes 21 problems covering topics such as: proving properties of analytic functions; finding radii of convergence of power series; properties of functions like the logarithm and tangent; mapping regions via Möbius transformations; and properties that characterize different types of Möbius transformations. The problems progress from simpler exercises to more complex proofs and geometric interpretations involving concepts like branches of the logarithm and conformal mappings.

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    tutorial of conway questions

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    This document provides a tutorial sheet for the course MTL 506: Complex Analysis. It includes 21 problems covering topics such as: proving properties of analytic functions; finding radii of convergence of power series; properties of functions like the logarithm and tangent; mapping regions via Möbius transformations; and properties that characterize different types of Möbius transformations. The problems progress from simpler exercises to more complex proofs and geometric interpretations involving concepts like branches of the logarithm and conformal mappings.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    47 views2 pages

    MTL 506 - Tutorial Sheet 3

    Uploaded by

    vamgadu
    This document provides a tutorial sheet for the course MTL 506: Complex Analysis. It includes 21 problems covering topics such as: proving properties of analytic functions; finding radii of convergence of power series; properties of functions like the logarithm and tangent; mapping regions via Möbius transformations; and properties that characterize different types of Möbius transformations. The problems progress from simpler exercises to more complex proofs and geometric interpretations involving concepts like branches of the logarithm and conformal mappings.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 2

    Department of Mathematics

    MTL 506: Complex Analysis


    Tutorial Sheet 3: Elementary Properties and Examples of Analytic
    Functions

    1. Prove Proposition 1.5

    2. Find the radius of convergence for each of the following power series:
    P 2
    (a) an z n , a C; (b) an z n , a C; (c) k n z n , k Z \ {0}; (d) z n! .
    P P P

    3. Show that the radius of convergence of the power series



    X (1)n n(n+1)
    z
    n=1
    n

    is 1, and discuss the convergence for z = 1, 1, and i.

    4. Show that f (z) = |z|2 = x2 + y 2 has a derivative only at the origin.

    5. Where is the function z 7 tan z = sin z/ cos z defined and where is it analytic?

    6. Suppose zn = rn ein and z = r ei C \ {z : z 0}, where n , (, ). Prove that if


    zn z, then n and rn r.

    7. Suppose that f : G C is a branch of the logarithm, and that n Z. Prove that z n =


    exp(nf (z)) for all z G.

    8. Show that the real part of the function z 1/2 is always positive.

    9. Let G = C \ {z : z 0}, and let n N. Find all analytic functions f : G C such that
    n
    z = f (z) for all z G.

    10. Suppose f : G C is analytic, and G is connected. Show that if f (z) R for all z G, then
    f is a constant.

    11. Let G be a region, and G? = {z C : z G}. If f : G C is analytic on G, prove that


    f ? : G? C defined by f ? (z) = f (z), is also analytic in G? .
    
    12. Find the image of z : <e z < 0, =m z < under the exponential function.
    az + b
    13. If T z = , show that T (C ) = R if and only if we can choose a, b, c, d to be real
    cz + d
    numbers.

    14. Show that if 2 , 3 , 4 are in , then eqn. (3.18) is satisfied if and only if z ? , 2 , 3 , 4 =
    
    
    z, 2 , 3 , 4 .

    15. Let D = {z : |z| < 1}. Find all Mbius transformations T such that T (D) = D.

    16. Suppose one circle is contained inside another, and that they are tangent at the point z = a.
    Map the region G between the two circles conformally onto the open unit disc.
    z ia
    17. Let < a < b < , and let T z = . Define the lines L1 = {z : =m z = b},
    z ib
    L2 = {z : =m z = a}, and L3 = {z : <e z = 0}. Pair the regions in Figure 1 with the regions
    in Figure 2 (see page 55).

    1
    18. Let T be as defined in # 17, and let log denote the principal branch of the logarithm.

    (a) Show that log(T z) is defined for all z except z = ic, a c b.


    
    (b) Show that h(z) = =m log(T z) = =m (log(z ia) log(z ib)), and that 0 < h(z) <
    for <e z > 0.

    (c) Show that log(z ic) is defined for <e z > 0 and c R, and that |=m log(z ic)| < 2
    if <e z > 0.
    (d) Show that
    Z b
    dt
    = i (log(z ib) log(z ia)) .
    a z it
    Conclude that
    Z b
    x ya yb
       
    h(x + iy) = 2 2
    dt = arctan arctan .
    a x + (y t) x x

    (e) Interpret the result in part (d) geometrically, and show that for <e z > 0, h(z) is the
    angle depicted in the figure on page 56.

    19. Let T be a Mbius transformation with fixed points z1 and z2 . If S is a Mbius transformation,
    then show that S 1 T S has fixed points S 1 z1 and S 1 z2 .

    20. (a) Show that a Mbius transformation has 0 and as its only fixed points if and only if
    it is a dilation.
    (b) Show that a Mbius transformation has as its only fixed points if and only if it is a
    translation.

    21. Let T be a Mbius transformation, T 6= identity. Show that a Mbius transformation S


    commutes with T if S and T have the same fixed points.

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