Singular Points, Laurent Series, and Residues

Abstract

This chapter is devoted to the systematic study of singular points, Laurent series expansion, and application of the theory of residues by the evaluation of integrals. We start with the result that reveals an important difference between the behavior of real and complex series—the theorem on the existence of at least one singular point on the circle of convergence.

Exercises

• 1. Find singular points and classify them:

  • (1) $\ \frac{1}{4z-z^{3} } \,$;

  • (2) $\ \frac{z^{4} }{1+z^{4} } \,$;

  • (3) $\ \frac{z+3}{z\left( z^{2} +25\right) ^{2} } \,$;

  • (4) $\ \frac{z^{5} }{\left( z+1\right) ^{3} } \,$;

  • (5) $\ \frac{e^{z} }{9+z^{2} } \,$;

  • (6) $\ \frac{9+z^{2} }{e^{z} } \,$;

  • (7) $\ \frac{1}{z^{3} \left( 4-\sin z\right) } \,$;

  • (8) $\ e^{\frac{z}{1-z} }$;

  • (9) $\ e^{z-\frac{1}{z} } \,$;

  • (10) $\ \frac{1}{\cos z} \,$;

  • (11) $\ \frac{\sin z}{z^{3} } \,$;

  • (12) $\ \sin \frac{1}{1+z} \,$;

  • (13) $\ \frac{1}{\sin z+\sin a} \,$;

  • (14) $\ e^{-z} \cos \frac{1}{z} \,$;

  • (15) $\ \frac{1-\cos z}{z^{2} } \,$;

  • (16) $\ \frac{1-\cos z}{\sin ^{2} z} \,$;

  • (17) $\ \frac{1}{z^{2} -1} \cos \frac{z\pi }{z+1} \,$;

  • (18) $\ \frac{1}{e^{z} -1} -\frac{1}{\sin z} \$.

• 2. Expand the function in the Laurent series at the given points and find the ring (or the disk) of convergence of the obtained series:

  • (1) $\ \frac{1}{z-5} \, ;\ z=0\, ,\ z=5\, ,\ z=\infty \,$;

  • (2) $\ \frac{2}{\left( z^{2} +1\right) ^{2} } \, ;\ z=0\, ,\ z=i\, ,\ z=\infty \,$;

  • (3) $\ z^{3} e^{\frac{1}{z} } \, ;\ z=0\, ,\ z=\infty \,$;

  • (4) $\ z^{2} \sin \frac{1}{z+i} \, ;\ z=-i\,$;

  • (5) $\ \cos \frac{z^{2} +2z}{\left( z+1\right) ^{2} } \, ;\ z=-1\,$;

  • (6) $\ e^{z+\frac{1}{z} } \, ;\ z=0\, ,\ z=\infty \,$.

• 3. Find the set of convergence of the Laurent series:

  • (1) $\ \sum _{n=-\infty }^{+\infty }2^{-\left| n\right| } z^{n}$;

  • (2) $\ \sum _{n=-\infty }^{+\infty }\frac{z^{3n} }{3^{n} +1} \,$;

  • (3) $\ \sum _{n=-\infty }^{+\infty }\frac{\left( z-1\right) ^{n} }{\mathrm{cosh\,} an} \, ,\ a>0\,$;

  • (4) $\ \sum _{n=-\infty }^{+\infty }\frac{z^{n} }{n^{2} +1} \,$;

  • (5) $\ \sum _{n=-\infty }^{+\infty }2^{-n^{2} } \left( z+1\right) ^{n}$;

  • (6) $\ \sum _{n=-\infty }^{+\infty }2^{-n^{2} } \left( z+1\right) ^{n^{2} }$;

  • (7) $\ \sum _{n=-\infty }^{+\infty }2^{n} z^{n} \$.

• 4. Find singular points and evaluate the residues at the single-valued isolated singularities:

  • (1) $\ \frac{z^{3} }{\left( z^{2} +4\right) ^{2} } \,$;

  • (2) $\ \frac{z^{5} }{\left( z^{2} +4\right) ^{2} } \,$;

  • (3) $\ \frac{e^{z} }{z^{3} \left( z^{2} +25\right) } \,$;

  • (4) $\ \frac{1}{\sin z} \,$;

  • (5) $\ \cos \frac{z}{z-1} \,$;

  • (6) $\ z^{3} \cos \frac{1}{z+1} \,$;

  • (7) $\ e^{z+\frac{1}{z} } \,$;

  • (8) $\ \frac{1}{z+z^{3} } \,$;

  • (9) $\ \frac{z^{2} }{1+z^{4} } \,$;

  • (10) $\ \frac{z^{2} }{\left( 1+z\right) ^{3} } \,$;

  • (11) $\ \frac{1}{\left( z^{2} +1\right) ^{3} } \,$;

  • (12) $\ \frac{z^{2n} }{\left( z-1\right) ^{n} } \, ,\, \, \forall n\in {\mathbb N}$;

  • (13) $\ \frac{1}{e^{z} +1} \,$;

  • (14) $\ \frac{\sin z\pi }{\left( z-1\right) ^{3} } \,$.

• 5. Let $z=a\ne \infty$ be an essential singularity of f(z) and a pole of g(z). Prove that a is essential singularity of $h(z)=f(z)g(z)$.

  Hint: use a proof by contradiction.

• 6. Assume that $z=a$ is a single-valued isolated singularity of f(z) and the function f(z) satisfies the condition $|f(z)|<M\cdot |z-a|^{-m}$ in a deleted neighborhood of a, where M and m are positive constants. Demonstrate that $z=a$ cannot be an essential singularity of f(z).

• 7. Let $z=a\ne \infty$ be a pole of order k of a function f(z). Determine what kind of point is $z=a$ for $f^{(n)}(z)$.

• 8. Suppose f(z) and g(z) have a pole of order m and n, respectively, at $z=\infty$. Prove that the composite function $F(z)=f(g(z))$ has a pole of order mn at $z=\infty$.

• 9. Let the Laurent series $\sum _{n=-\infty }^{+\infty } c_n (z-a)^n$ be convergent in the closed ring $\overline{D}=\{0<r\le |z-a|\le R\}$. Demonstrate that $|c_n|\le M (r^{-n}+R^{-n})$, $\forall n\in \mathbb {Z}$, where M is a constant independent of n.

• 10. Let $z=0$ and $z=\infty$ be single-valued isolated singularities (or regular points) of a function f(z). Prove that if f(z) is an even function, then $\mathop {{\text {res}}}\limits _{z=0} f(z)=\mathop {{\text {res}}}\limits _{z=\infty } f(z)=0$.

• 11. Let $z=a$ and $z=-a$ be single-valued isolated singularities of f(z). Demonstrate that:

  • (1) if f(z) is an even function, then $\mathop {{\text {res}}}\limits _{z=a} f(z)=-\mathop {{\text {res}}}\limits _{z=-a} f(z)$;

  • (2) if f(z) is an odd function, then $\mathop {{\text {res}}}\limits _{z=a} f(z)=\mathop {{\text {res}}}\limits _{z=-a} f(z)$.

• 12. Suppose that f(z) and g(z) are analytic at $z=a\ne \infty$ and have a zero of order m at this point. Demonstrate that:

  • (1) $\mathop {{\text {res}}}\limits _{z=a} \left( \frac{f(z)}{g(z)}\cdot \frac{1}{z-a}\right) =\frac{f^{(m)}(a)}{g^{(m)}(a)}$;

  • (2) $\mathop {{\text {res}}}\limits _{z=a} \left( \frac{f(z)}{g(z)}\cdot \frac{1}{(z-a)^2}\right) =\frac{1}{m+1}\frac{f^{(m)}(a)}{g^{(m)}(a)} \left( \frac{f^{(m+1)}(a)}{f^{(m)}(a)}-\frac{g^{(m+1)}(a)}{g^{(m)}(a)} \right)$.

• 13. Assume that a function g(z) is analytic at a, $g'(a)\ne 0$, while a function f(z) has a simple pole at $b=g(a)$ and $\mathop {{\text {res}}}\limits _{z=b} f(z)=B$. Find $\mathop {{\text {res}}}\limits _{z=a} f(g(z))$.

• 14. Assume that f(z) and g(z) are analytic at $z=a\ne \infty$, $f(a)\ne 0$ and g(z) has a zero of order 2 at a. Find $\mathop {{\text {res}}}\limits _{z=a} \frac{f(z)}{g(z)}$.

  Hint: $\mathop {{\text {res}}}\limits _{z=a} \frac{f(z)}{g(z)}=\frac{2f'(a)}{g''(a)}-\frac{2}{3}\frac{f(a)g'''(a)}{(g''(a))^2}$.

• 15. Suppose that $z=\infty$ is a single-valued isolated singularity of f(z), that is, in a deleted neighborhood of $\infty$ the function f(z) has a representation in the Laurent series: $f(z)=\sum _{n=-\infty }^{+\infty } c_nz^n$. Find $\mathop {{\text {res}}}\limits _{z=\infty } f^2(z)$ if:

  • (1) $z=\infty$ is a removable singularity;

  • (2) $z=\infty$ is a pole of order k;

  • (3) $z=\infty$ is an essential singularity.

• 16. Find $\mathop {{\text {res}}}\limits _{z=a} \left( g(z)\frac{f'(z)}{f(z)}\right)$ if g(z) is analytic at $z=a\ne \infty$ and f(z):

  • (1) has a zero of order m at $z=a$;

  • (2) has a pole of order m at $z=a$.

    Hint: (1) $\mathop {{\text {res}}}\limits _{z=a} \left( g(z)\frac{f'(z)}{f(z)}\right) =mg(a)$; (2) $\mathop {{\text {res}}}\limits _{z=a} \left( g(z)\frac{f'(z)}{f(z)}\right) =-mg(a)$.

• 17. Suppose that D is a finite domain, g(z) is analytic in D and continuous in $\overline{D}$, and f(z) is analytic in D and continuous in $\overline{D}$ except at a finite number of poles $b_j\in D$, $j=1,\ldots , m$ of order $k_j$, respectively, and additionally $f(z)\ne 0$, $\forall z\in \partial D$. Show that if f(z) has a finite number of zeros $a_k\in D$, $k=1,\ldots , n$ of order $l_k$, respectively, then

  $$\begin{aligned} \int _{\partial D} g(z)\frac{f'(z)}{f(z)}dz=2\pi i \sum _{k=1}^{n} l_k g(a_k)- 2\pi i \sum _{j=1}^{m} k_j g(b_j) . \end{aligned}$$

  Hint: make use of Exercise 16.

18$^*$.:

Let f(z) be an entire function. Prove that any analytic branch of the function $\ln \frac{z-b}{z-a}$, which is analytic in a neighborhood of $\infty$, satisfies the formula $\mathop {{\text {res}}}\limits _{z=\infty } \left( f(z)\ln \frac{z-b}{z-a} \right) =\int _a^b f(z)dz$.

Hint: find the Laurent series expansion at $\infty$ of an analytic branch of $f(z)\ln \frac{z-b}{z-a}$.

19.:

Evaluate integrals using the theory of residues:

• (1) $\ \oint _{x^{2} +y^{2} =2x}\frac{dz}{z^{4} +1} \,$;

• (2) $\, \oint _{\left| z-2\right| =1}\frac{\left( z-1\right) dz}{\left( z+1\right) \left( z-2\right) ^{2} } \,$;

• (3) $\, \oint _{\left| z\right| =2}\frac{zdz}{\left( z-5\right) \left( z^{5} -1\right) } \,$;

• (4) $\, \oint _{\left| z\right| =3}\frac{z^{3} dz}{z^{4} -1} \,$;

• (5) $\, \oint _{\left| z\right| =2}\sin \frac{1}{z} dz \,$;

• (6) $\, \oint _{\left| z\right| =2} \sin ^{2} \frac{1}{z} dz \,$;

• (7) $\, \oint _{\left| z\right| =3}\left( z^{2} -2\right) \sin \frac{1}{z-2} dz \,$;

• (8) $\, \oint _{\left| z\right| =1}z^{n} e^{\frac{3}{z} } dz,\ \forall n\in {\mathbb Z}\,$;

• (9) $\, \oint _{\left| z\right| =4}\frac{z}{z+3} e^{\frac{1}{3z} } dz \,$;

• (10) $\, \oint _{\left| z\right| =2}\frac{dz}{z^{3} \left( z^{10} -2\right) } \,$;

• (11) $\, \oint _{\left| z\right| =3}\frac{z^{2} }{\left( z-1\right) \left( z-2\right) } \sin ^{2} \frac{1}{z} \, dz \,$;

• (12) $\, \oint _{\left| z\right| =5}\left( z^{2} +1\right) e^{\frac{1}{z-1} } dz \,$;

• (13) $\, \oint _{\left| z\right| =2}\left( z^{2} +5z+3\right) \sin \frac{1}{z} dz \,$;

• (14) $\ \int _{0}^{2\pi }\frac{d\varphi }{\left( a+b\cos \varphi \right) ^{2} } \, \, ,\ a>b>0\,$;

• (15) $\ \int _{-\pi }^{\pi }\frac{1+4\cos ^{2} \varphi }{\left( 17-8\cos \varphi \right) ^{2} } \, d\varphi$;

• (16) $\ \int _{0}^{2\pi }\frac{2+3i \sin 2\varphi }{15-8i\, \sin \varphi } d\varphi$;

(17$^*$):

$\, \int _{0}^{2\pi }e^{\cos \varphi } \cos \left( n\varphi -\sin \varphi \right) d\varphi ,\, \, \forall n\in {\mathbb Z}$;

• (18) $\ \int _{0}^{2\pi }\frac{\cos ^{2} \varphi }{13+12\cos \varphi } d\varphi \,$;

• (19) $\ \int _{0}^{\pi }\frac{\cos ^{4} \varphi }{1+\sin ^{2} \varphi } d\varphi \,$;

20$^*$):

$\, \int _{-\pi }^{\pi }\frac{\sin n\varphi \, d\varphi }{1\!-\!2a\sin \varphi \!+\!a^{2} } \, ,\ -1\!<\!a\!<\!1,\, a\!\ne \!0, \, \forall n\in {\mathbb N}$;

• (21) $\, \int _{-\infty }^{+\infty }\!\frac{x\, dx}{\left( x^{2} \!+\!4x\!+\!13\right) ^{2} } \,$;

• (22) $\ \int _{0}^{+\infty }\frac{dx}{\left( x^{2} +1\right) ^{n} },\ \forall n\in {\mathbb N}$;

• (23) $\ \int _{0}^{+\infty }\frac{x^{2} +7}{x^{4} +10x^{2} +9} dx\,$;

• (24) $\ \int _{-\infty }^{+\infty }\frac{x^{2} -3x+1}{\left( x^{2} +4x+8\right) ^{2} } dx \,$;

• (25) $\ \int _{0}^{+\infty }\frac{x^{2} +9}{x^{6} +1} dx \,$;

• (26) $\ \int _{-\infty }^{+\infty }\frac{\left( x-1\right) e^{ix} }{x^{2} -2x+2} dx \,$;

• (27) $\, \int _{-\infty }^{+\infty }\! \frac{e^{ix} dx}{\left( x^{2}\! +4ix\!-\!5\right) ^{2} } \,$;

• (28) $\, \int _{-\infty }^{+\infty }\!\! \frac{\left( x\!+\!1\right) e^{-3ix} }{x^{2} \!-2x\!+5} dx\,$;

• (29) $\, \int _{-\infty }^{+\infty }\!\!\! \frac{x\cos x}{x^{2} \!+4x\!+20} dx\,$;

• (30) $\ \int _{-\infty }^{+\infty }\frac{\left( x+1\right) \sin 3x}{\left( x^{2} +6x+10\right) ^{2} } dx\,$;

• (31) $\ \int _{-\infty }^{+\infty }\frac{ \sin 2x\, dx}{\left( x^{2} -7x+10\right) \left( x^{2} +1\right) } \,$;

• (32) $\ \int _{-\infty }^{+\infty }\frac{\cos 3x}{\left( x-1\right) \left( x^{2} +1\right) ^{2} } dx \,$.

20$^*$.:

Let f(z) be analytic in a ring $R<|z|<+\infty$ and $\mathop {{\text {res}}}\limits _{z=\infty } f(z)=0$. Prove that in this ring there exists an analytic primitive of the function f(z).

Hint: show that the integral $\int _a^{z} f(\zeta ) d\zeta$, $R<|a|, |z|<+\infty$ does not depend on the form of curve lying in the given ring and connecting the points a and z.

• 21. Find the number of roots of the given equation in the disk $\left| z\right| <1$ using Rouché’s Theorem:

  • (1) $\ 3z^{6} -z^{4} +2z^{2} +z-8=0\,$;

  • (2) $\ z^{5} -6z^{4} +z^{3} -2z+1=0\,$;

  • (3) $\ 2z^{4} -5z+2=0\,$;

  • (4) $\ z^{7} -5z^{4} +z^{2} -2=0\,$;

  • (5) $\ z^{8} -4z^{5} +z^{2} -1=0\,$.

• 22. Find the number of roots of the equation $z^{4} +7z-1=0$:

  • (1) in the disk $\left| z\right| <1$;

  • (2) in the ring $1<\left| z\right| <2$.

• 23. Show that the equation $ze^{\lambda -z} =1$, where $\lambda >1$, has exactly one root in the disk $\left| z\right| <1$ and this root is real.

• 24. Show that the equation $z=\lambda -e^{-z}$, where $\lambda >1$, has the only root in the right half-plane $\mathrm{Re \,}z>0$ and this root is real.

• 25. Verify whether the statement of Rouché’s theorem will be true if the inequality $|f(z)|<|F(z)|$ on the boundary of a domain D is replaced by the non-strict version $|f(z)|\le |F(z)|$.

  Hint: find a counterexample.

• 26. Let $D=\{|z|<1\}$. Consider the equation $e^z=az^n$. Prove that:

  • (1) if $|a|>e$, then the equation has n roots in D;

  • (2) if $|a|<e^{-1}$, then the equation has no roots in D.

27$^*$.:

Prove that the equation $z\sin z=1$ has a countable set of roots in the complex plane and all these roots are real.

Hint: find the number of real roots of the equation $x\sin x=1$ in the segment $[-R,R]$ and compare with the number of roots of the given equation in the square $\overline{D}=\{ x\in [-R,R], y\in [-R,R] \}$, where $R=\left( n+\frac{1}{2}\right) \pi$, $\forall n\in \mathbb {N}$.