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1. Find the Laurent expansion about \(a=0\) which represents the following functions in the indicated regions.
1. \(\frac{z^{2}-1}{(z+2)(z+3)}\) for \(|z|>3\).
\[\text{Hint:}\quad\frac{z^{2}-1}{(z+2)(z+3)}=1-\frac{5z+7}{(z+2)(z+3)}=1+\frac {A}{z+2}+\frac{B}{z+3}\]
with \(A=3\), \(B=-8\).
* \(\frac{24}{z^{2}(z-1)(z+2)}\) for \(0<|z|<1\). \[\mbox{Hint}:\quad\frac{24}{z^{2}(z-1)(z+2)}=\frac{A}{z}+\frac{B}{z^{2}}+\frac{C} {z-1}+\frac{D}{z+2}\] where \(A=-6\), \(B=-12\), \(C=8\), \(D=-2\).
2. Find the Laurent series about \(a=1\) and then \(a=0\) for \(f(z)=\frac{1}{z^{2}-z^{3}}\).
3. Find the principal part and residue of \(f(z)=\frac{z^{3}+z^{2}}{(z-1)^{2}}\) at \(a=1\).
Hint: Expand \(z^{3}+z^{2}\) in powers of \(z-1\): \(z^{3}+z^{2}=2+5(z-1)+4(z-1)^{2}+(z-1)^{3}\).
4. For \(f(z)=e^{\frac{1}{z^{2}}}\), for \(|z|>0\), show that \(a=0\) is an isolated essential singularity.
2. For \(f(z)=\frac{\sin z}{z^{4}}\), \(|z|>0\), show that \(a=0\) is a pole of order 3.
5. Locate and classify all singularities of the following functions:
* \(f(z)=\frac{2}{(z-3)^{2}}+\frac{1}{z-3}+e^{z}\).
* \(f(z)=\sin z+\sin\frac{1}{z}\).
* \(f(z)=\frac{\cos z}{z-\frac{\pi}{2}}\).
6. Show that the function \(f(z)=\frac{z^{3}-8}{(z-2)^{2}(z+i)^{4}}\) has a simple pole at 2 and a pole of order 4 at \(-i\).
7. Let \(f\) be holomorphic on \(\mathbb{C}\setminus\{0\}\). Show that the Laurent expansions for \(f\) valid in the regions \(\{z:|z|>0\}\) and \(\{z:|z|>1\}\) are the same.
8. Find the Laurent series for \(f(z)=\frac{8z+1}{z(1-z)}\) valid for \(0<|z|<1\).
9. Let \(f\) and \(g\) be continuous on \(\overline{A}\) and holomorphic on \(A\), where \(A\) is open, i.e., a connected and bounded region. If \(f=g\) on \(\partial A\), show that \(f=g\) on all of \(\overline{A}\).
10. If \(f\) is entire and bounded on the real axis, then \(f\) is constant. Prove or give a counterexample.
11. Find the Laurent series for \[f(z)=\frac{4}{(1-z)(z+3)}\] in the annulus \(\{z:\ 1<|z|<3\}\).
12. Find the residue of the following functions at \(z=0\)
1. \(f(z)=\frac{z^{2}+1}{z}\),
2. \(f(z)=\frac{\sin z}{z^{4}}\).
13. Evaluate:
1. \(\int_{\gamma}\operatorname{Re}zdz\) where \(\gamma(t)=|z|=1\),
2. \(\int_{0}^{3+i}\sin zdz\),
3. \(\int_{\gamma}\frac{z+4}{z^{4}+2iz^{3}}\,dz\) where \(\gamma\) is the circle \(|z|=1\).
14. Use the residue theorem to evaluate the following:
1. \(\int_{\gamma}e^{4/z-2}\ dz\) where \(\gamma\) is \(|z-1|=3\),
2. \(\int_{\gamma}\frac{e^{z}}{z^{3}+2z^{2}}\,dz\) where \(\gamma\) is \(|z|=3\),
3. \(\int_{0}^{2\pi}\frac{1}{10-6\text{cos}\theta}\ d\theta\).
15. Let \(f,g\) be holomorphic functions on \(D(a;r)\). Assume that \(f\) has a zero of order \(m\) while \(g\) has a zero of order \(m+1\) at \(a\). Show that
\[\operatorname{Res}\left(\frac{f(z)}{g(z)};a\right)=(m+1)\frac{f^{m}(a)}{g^{(m +1)}(a)}.\]
16. Evaluate \(\int_{\gamma}\tan z\,dz\) where \(\gamma=|z|=2\).
17. Let \(f\) be holomorphic inside and on a positively oriented contour \(\gamma\) except at the point \(a\) inside \(\gamma\), where it has a pole of order \(m\). Let \(\sum_{n=-m}^{\infty}c_{n}(z-a)^{n}\) be the Laurent expansion of \(f\) about \(a\). Show that \(\int_{\gamma}f(z)dz=2\pi ic_{-1}\).
18. Find the Laurent series that converges in the annulus \(1<|z|<2\) to a branch of the function \(\log\left(\frac{z(2-z)}{1-z}\right)\).
19. Show that \(\int_{-\infty}^{\infty}\frac{e^{ax}}{1+e^{x}}\,dx=\frac{\pi}{ \text{sin}ax}\) for \(\,0 Hint: Consider \(f(z)=\frac{e^{az}}{1+e^{z}}\) use residue theorem over the contour \[\gamma=[-R,\,R]\cup\gamma_{1}\cup\gamma_{2}\cup\gamma_{3}\] as shown in the following Figure 3.44. ### 3.10 The Bieberbach Conjecture One of the most celebrated conjectures in classical analysis which stood as a challenge to mathematicians for nearly 70 years is called the Bieberbach conjecture. This conjecture appeared in a footnote to a paper [11] of a German mathematician, Ludwig Bieberbach, in 1916 and was solved by Louis de Branges of Purdue University in 1984 [19]. The Bieberbach conjecture is appealing partly because it is simple to pose, and it states that under reasonable restrictions the coefficients of a power series are not too large. The Bieberbach conjecture concerns functions which are both holomorphic and also univalent. A holomorphic function is _univalent_ if it is one-to-one (\(f(z_{1})\neq f(z_{2})\) unless \(z_{1}=z_{2}\)). Univalent functions have many interesting properties [20], thus we may wonder if we can say anything about the coefficients in their Taylor expansions. It turns out that if we also assume \(f(0)=0\) and \(f^{\prime}(0)=1\) then the Taylor series for \(f\) takes the form \[f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots\] Figure 3.44: Rectangular contour with complex coefficients \(a_{2},a_{3},\dots\). We use the letter \(S\) (for Schlicht) for the class of univalent and holomorphic functions \(f:\mathbb{D}\to\mathbb{C}\) which satisfy the normalization conditions \(f(0)=0\) and \(f^{\prime}(0)=1\). **Bieberbach Conjecture** : For each \(f\in S\), \(|a_{n}|\leq n\quad\text{for}\quad n=2,3,\dots\) The inequality is strict for every \(n\) unless \(f\) is a rotation of the Koebe function \(k(z)\) where \[k(z)=\sum_{n=1}^{\infty}nz^{n}=z+2z^{2}+3z^{3}+\cdots.\] The principal result of Bieberbach’s original paper was the second coefficient theorem, \(|a_{2}|\leq 2\), and that equality holds for the Koebe function. Before de Branges’ general proof of \(|a_{n}|\leq n\), this conjecture was known to be true only for \(n\leq 6\). There are several books and papers written on this conjecture; we refer the reader to [24, 55] for more detailed information. The Koebe function \(k(z)\) mentioned in this conjecture is \[k(z)=\frac{z}{(1-z)^{2}}=z\,\frac{d}{dz}\left[\frac{1}{(1-z)}\right]=z+2z^{2} +3z^{3}+\cdots\] which converges for every \(z\) in the disc \(|z|<1\). To see why \(k(z)\) is univalent on the disc and to find its image, consider \(k(z)\) in the following form: \[k(z)=\frac{1}{4}\left[\left(\frac{1+z}{1-z}\right)^{2}-1\right].\] We see that \(k(z)\) is composition of the following mappings \[p=\frac{1+z}{1-z},\qquad q=p^{2},\qquad w=\frac{1}{4}(q-1)\] First \(p\) is a linear fractional transformation that maps the unit disc univalently onto the right half of the \(p\)-plane. The mapping \(q=p^{2}\) is one-to-one when restricted to the right half-plane; its image is the entire \(q\) plane minus the non-negative real axis. Finally, the last mapping \(w\) is a simple translation followed by a dilation with a factor of \(\frac{1}{4}\) as shown in Figure 3.45. Given a function \(f\) satisfying \(f(0)=0,\ f^{\prime}(0)=1\) and a real number \(\alpha\), then \[g(z)=e^{-i\alpha}f(e^{i\alpha}z)\]
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