Isolated Singularities and Residues

Using residue calculus evaluate

Use residue calculus to show that \[ \int _0^\infty \frac {\log x}{x^3-1}\,dx=\frac {4\pi ^2}{27}. \] Hint: Use an appropriate branch of the function \((\log z)/(z^3-1)\) and a sector of aperture \(2\pi /3\).

State and prove Riemann’s theorem on removable singularities.

1. (10 pts) Compute the Laurent series expansion about the point \(z=0\) for \[ f(z)=\frac {i+2}{(z+i)(z-2)}, \] in the region \(1<|z|<2\).
2. (10 pts) Discuss the type of singularity (removable, pole and order, essential, branch, cluster). If the type is a pole give the strength of the pole, and give the nature (isolated or not). Include the point at infinity. \[ f(z)=\frac {e^{2z}-1}{z} \]

Prove the argument principle: Assume that \(f\) is a meromorphic function inside and on a simple closed contour \(C\) (oriented in the typical counter-clockwise fashion), that \(f\) has exactly \(N\) zeros (counting multiplicity) in this region all lying within \(C\), and that \(f\) has exactly \(M\) poles (counting order), all lying within \(C\). Prove that \[ \frac {1}{2\pi i}\oint _C \frac {f'(z)}{f(z)}\,dz=N-M. \]

Evaluate the following integral by residue integration: \[ \int _0^\infty \frac {dx}{x^3+a^3},\qquad a>0. \] Clearly indicate the contour of integration and the treatment of the integral along each segment.

Complete the following:

1. (5 pts) Compute the Laurent series expansion about the point \(z=0\) for \[ f(z)=\frac {1}{(z-i)(z+2)}, \] in the region \(1<|z|<2\).
2. (5 pts) Describe the singularities of the function \[ f(z)=\frac {z+1}{z\sin z}. \]

Find all singularities of the function \(\frac {1}{e^z+1}\) in \(\mathbb {C}\). If they are poles, compute the residue.

Let \(f\) be holomorphic on \(\mathbb {C}\setminus \{0,2\}\) with simple poles at 0 and 2. Suppose that \(\int _{|z|=1} f(z)\,dz=2\pi i\) and \(\int _{|z|=3} f(z)\,dz=0\). Also suppose that \(f\) is bounded on \(|z| > 3\). List all such functions (with a proof that the list is complete).

Let \(a, b > 0\). Using the residue calculus evaluate the improper integral

Suppose \(f:\mathbb {D}\setminus \{0\}\to \mathbb {C}\) is holomorphic and has an essential singularity at 0. Show that for all \(\epsilon > 0\) there is \(z\) near 0 so that \[ \left |f(z)-\frac {1}{z^4}\right |\le \epsilon . \]

Using the residue calculus evaluate the integral

Show that if \(f,g:B(0,1)\setminus \{0\}\to \mathbb {C}\) are holomorphic, \(f\) has an essential singularity at 0 and \(g\) has a pole at 0 then \(\frac {g}{f}\) has an essential singularity at 0.

Show that any function which is meromorphic on the Riemann sphere, \(\widehat {\mathbb {C}}=\mathbb {C}\cup \{\infty \}\), is either constant or can be written as the sum of the singular parts of its Laurent series at its poles, in particular conclude that the function is rational.

Show that if \(f:\{z:0<|z|<1\}\to \mathbb {C}\) is meromorphic and has poles at \(\frac {1}{2},\frac {1}{3},\frac {1}{4},\ldots \) then the range of \(f\) is dense in \(\mathbb {C}\).

Let \(C([0,1])\) be the Banach space of continuous functions on \([0,1]\) with the sup norm and view it as a natural subspace of \(L^\infty ([0,1])\), where \([0,1]\) is given Lebesgue measure, and the norm is essential supremum. Prove the existence of a bounded functional on \(L^\infty ([0,1])\) which is not identically \(0\) but vanishes on \(C([0,1])\).

Evaluate the integral by the method of residues: \[ \int _0^\infty \frac {x^{1/3}}{1+x^2}\,dx. \]

1. Find the Laurent expansion of \[ f(z)=\frac {z}{(z+i)(z-2)} \] for \(1<\lvert z\rvert <2\).
2. Classify the singularities of the function \[ f(z)=\frac {z+1}{z\sin (z)}. \]

Let \((X,\mathcal M,\mu )\) be a measure space. Assume there exists \(\delta >0\) such that for every \(E\in \mathcal M\), either \(\mu (E)=0\) or \(\mu (E)>\delta \). Let \(f\in L^1(X)\). Prove that \(f\) is essentially bounded.

Show that all the singularities of the function \(f(z)=1/(e^z+3z)\) have order \(1\). You do not need to find the singularities.

Let \(f\) be a nonconstant entire function. Assume that the set \[ \{z\in \mathbb C:|f(z)|<1\} \] is unbounded. Show that \(f\) has an essential singularity at \(\infty \).

Determine the poles and their orders of the function \[ \frac {1}{e^z-1}-\frac 1z. \]

Prove that if all singularities of an analytic function \(f(z)\) in the extended complex plane \(\overline {\mathbb {C}}\) are poles, then \(f(z)\) is a rational function (i.e. the ratio of two polynomials).

Evaluate the integral by the method of residues: \[ \int _0^\infty \frac {x^2}{x^4+5x^2+6}\,dx. \]

Evaluate the integral by the method of residues: \[ \int _0^\infty \frac {x\sin x}{x^2+a^2}\,dx, \] for \(a\in \mathbb R\).

Let \(f(z)\) be analytic in a deleted neighborhood of \(z=a\) and \[ \lim _{z\to a}(z-a)f(z)=0. \] Does it mean that \(f(z)\) has a removable singularity at \(a\)? If so, prove it. If not, give a counterexample.

Let \(\wp (z)\) be the Weierstrass \(\wp \) function. Show that any even elliptic function \(f(z)\) with periods \(\omega _1,\omega _2\) can be expressed in the form \[ (\text {Constant})\prod _{k=1}^{n} \frac {\wp (z)-\wp (a_k)}{\wp (z)-\wp (b_k)} \] for some Constant, \(a_k,b_k\in \mathbb C\), provided that \(0\) is neither a zero nor a pole. What is the corresponding form if \(f(z)\) either vanishes or has poles at \(z=0\)?

Using the residue theorem, find the integral \[ \int _{-\infty }^{\infty }\frac {\sin x}{x^2+4x+20}\,dx. \]

1. Show that any analytic function (not identically equal to zero) can have only isolated zeros inside its analyticity domain.
2. Can an analytic function have a non-isolated singularuty?
3. Prove the Uniqueness Theorem: If two functions are analytic in a domain \(D\) and equal on some set of points that has a limiting point inside \(D\), then these functions are identically equal in \(D\).

Let \(f\) be a nonzero entire function. Assume that the set \(Z(f)=\{z\in \mathbb C:\ f(z)=0\}\) is unbounded. Show that \(f\) had an essential singularity at \(\infty \).

1. Show that any analytic function (not identically equal to zero) can have only isolated zeros inside its analyticity domain.
2. Prove the Uniqueness Theorem: If two functions are analytic in a domain \(D\) and equal on some set of points that has a limiting point inside \(D\), then these functions are identically equal in \(D\).

The function \(f(z)\) is analytic in the extended complex plane except for two points \(z=0\) and \(z=2\), where it has simple poles. It is known that \[ \oint _{|z|=1} f(z)\,dz=1, \qquad \oint _{|z|=3} f(z)\,dz=0. \] Find \(f(z)\) (unique up to an arbitrary additive constant).

Let \(f\) and \(g\) be analytic functions defined on an open set \(U\subset \mathbb C\). Assume that \(g\) is not identically equal to zero. Prove that \(F=f/g\) does not have essential singularities in \(U\).

Let \(f\) be a holomorphic function and \(a\) an isolated singularity of \(f\). Prove that the following statements are equivalent:

(a) there exists a disk \(D(a,\epsilon )=\{z\in \mathbb C\mid |z-a|<\epsilon \}\) and a constant \(C>0\) such that \[ |f(z)|\ge \frac {C}{|z-a|^n} \] for all \(z\in D(a,\epsilon )-\{a\}\);

(b) \(f\) has a pole of order \(m\ge n\) at \(a\).

Find the Laurent series of the function \[ f(z)=\frac {1}{(z-z^2)^2} \] at \(z=0\).

(a) What kind of isolated singularity does \(f\) have at \(0\)?

(b) What is the residue of \(f\) at \(0\)?

1. Formulate and derive the argument principle [which determines the difference between the number of zeros \((N_0)\) and poles \((N_\infty )\) of an analytic function \(f(z)\)].
2. Applying this principle, determine the number of zeros located inside the first quadrant \(\{z=x+iy:\ x>0,\ y>0\}\) of the function \(f(z)=z^5+1\).

Find all isolated singularities of the function \[ f(z)=\sin (z)\sin \left (\frac {1}{z}\right ) \] in \(\mathbb C\). Determine the residues of \(f\) at these isolated singularities.

Evaluate the integral \[ \int _{-\infty }^{\infty }\frac {x\sin x}{x^2+2x+10}\,dx \] using the residue theorem.

Let \[ f(z)=\sin \left (\frac {z}{z+1}\right ). \] (i) Determine all isolated singularities of \(f\) and their type;

(ii) Find the Laurent expansions of \(f\) at these singularities;

(iii) Find the residues of \(f\) at these singularities.

Evaluate the integral \[ \int _{-\infty }^{\infty }\frac {x\cos x}{x^2-2x+10}\,dx \] using the residue theorem.

Use antiderivatives to evaluate the contour integral \[ I=\int _C\frac {1}{z+i} \] where \(C\) is a contour (possibly with loops) starting at \(z=0\) and ending at \(z=1\). Express your answer in the form \(I=a+ib\) where \(a\) and \(b\) are real. The answer is multivalued. Explain the interpretation of the different values, and account for the difference between successive values using the residue theorem.

Let \[ f(z)=\frac {6z+2}{(z^2-1)(z+3)}. \]

(a) Use the residue theorem to compute \(\int _\gamma f(z)\,dz\), where \(\gamma \) is the positively oriented circle \(|z|=2\).

(b) Compute the Laurent series for \(f(z)\) which converges in the annulus \(1<|z|<3\). Use this to check your computation of the integral in (a).

Suppose \(f(z)\) is analytic in some neighborhood of the point \(z_0\) without the point \(z_0\) itself; let \(z_0\) be an essential singularity of \(f(z)\).

Show that for any complex number \(C\), there is a sequence of points \(z_n\) (\(n=0,1,2,\ldots \)) such that \[ z_n\to z_0\qquad \text {and}\qquad f(z_n)\to C. \] (In other words, in every neighborhood of an essential singularity, the function \(f(z)\) is arbitrary close to every complex number.)

Suppose \(a\) is a real number greater than \(1\). Compute \[ \int _0^{2\pi }\frac {d\theta }{a+\sin \theta } \] using residues. Hint: \(\sin \theta =\frac {z-z^{-1}}{2i}\) where \(z=e^{i\theta }\).

1. Use contour integration to explicitly evaluate \[ f(a)=\int _0^\infty \frac {x\sin x\,dx}{x^2-a} \] for complex \(a\). For what values of complex \(a\) is this possible?
2. Is the function \(f(a)\) an analytic function of \(a\)? Identify all singularities, branch points, branch cuts, etc.

Let \(a\) be an isolated singularity of the meromorphic function \(f:\Omega \to \mathbb C\). Prove that if \(a\) is an essential singularity, then in any neighborhood of \(a\), the function \(f\) takes values arbitrarily close to any complex number.

Let \(f\) be a function holomorphic on \(D\) and continuous on \(\overline {D}\). Assume that \(|f(z)|=1\) whenever \(|z|=1\). Show that \(f\) can be extended to a meromorphic function on the whole \(\mathbb C\), with at most finitely many poles.

(Hint: Starting with the Schwarz reflection principle for the upper half plane, deduce that an appropriate reflection continuation that can be used in this setting is \(z\mapsto \frac {1}{f(1/\bar z)}\).)

Let \(\Delta \) be the open unit disk and \(\gamma \) the unit circle in \(\mathbb C\) and \(\phi :\gamma \to \mathbb C\) a continuous function. Let \(g\) be a meromorphic function on \(\mathbb C\) that has a single simple pole at \(0\). Define a function \(f:\Delta \to \mathbb C\) by the formula \[ f(z)=\int _\gamma \phi (w)g(w-z)\,dw. \] Show that \(f\) is holomorphic.

Let \[ f(z)=\frac {e^{\frac {1}{z-1}}}{e^z-1}. \] Determine all isolated singularities of \(f\) and their type. Find the residues of \(f\) at its poles.

Let \(f\) be a holomorphic function with isolated singularity at point \(a\). Assume that \(g=\frac {1}{f}\) also has an isolated singularity at \(a\), and that \(a\) is not a removable singularity for either \(f\) or \(g\). Determine what type of isolated singularity \(a\) is for \(f\) and \(g\).

Do you know an example of such a function \(f\)?

Let \(f\) be a function holomorphic in \(\mathbb C-\{0,2\}\). Assume that:

(i) \(0\) and \(2\) are first order poles of \(f\);

(ii) that \(f\) is bounded for \(|z|\ge R\) for sufficiently large \(R\);

(iii) the integral \(\int _\gamma f(z)\,dz\) is equal to:

a) \(2\pi i\) if \(\gamma \) is a positively oriented circle of radius \(1\) centered at \(0\);

b) \(0\) if \(\gamma \) is a positively oriented circle of radius \(3\) centered at \(0\).

Determine the function \(f\).

Fix positive integers \(k\) and \(l\). Describe the space of meromorphic functions on the Riemann sphere that have a have a pole of order at most \(k\) at \(\infty \) and a zero of order at least \(l\) at the origin.

Determine the poles and their orders of the function \[ \frac {1}{e^z-1}-\frac {1}{z}. \]

Let \[ f(z)=e^{\frac {z}{z-1}}. \] Determine all isolated singularities of \(f\) and their type. Find the residues of \(f\) at these isolated singularities.

Suppose \(a\) is a real number greater than \(1\). Compute \[ \int _0^{2\pi }\frac {d\theta }{a+\sin \theta } \] using residues. Hint: \(\sin \theta =\frac {z-z^{-1}}{2i}\) where \(z=e^{i\theta }\).

Determine the poles and their orders of the function \[ \frac {1}{e^z-1}-\frac {1}{z}. \]

Let \(f\) be an injective entire function. Put \(g(z)=f(1/z)\). Show that \(g\) does not have an essential singularity at \(0\). In particular, it follows that \(f(z)=az+b\). Why?

Prove that the Hermite polynomials form a complete set on \(-\infty <x<\infty \) with the inner product \(\langle u,v\rangle =\int _{-\infty }^{\infty }u(x)v(x)e^{-x^2/2}\,dx\).

Hint: The following steps will prove useful:

1. Show that if the Hermite-Bessel coefficients of \(f(x)\) are all zero, then \(\int _{-\infty }^{\infty }e^{-x^2/2}x^nf(x)\,dx=0\) for all \(n\).
2. Show that the Fourier transform \(F(z)\) of the function \(f(x)e^{-x^2/2}\) has no singularities in the complex plane and is therefore an entire function.
3. Calculate the Taylor series of \(F(z)\) and show that it is zero. What conclusion can you draw about \(f(x)\) and what does this say about Hermite polynomials?

(a) Let \(f\) be an entire function. Assume that there exists a polynomial \(g\) such that for each \(z\in \mathbb C\), \(|f(z)|\le |g(z)|\). Show that \(f\) is also a polynomial.

(b) Can we draw the same conclusion if the function \(g\) is rational? In other words, let \(f\) be an entire function and \(g\) a rational function such that for each \(z\in \mathbb C\), which is not a pole of \(g\), \(|f(z)|\le |g(z)|\). Does it follow that \(f\) is a polynomial?

1. State and prove Jordan’s Lemma of complex function theory.
2. Use Jordan’s Lemma and the residue theorem to prove the equation \[ \int _0^\infty \frac {\cos (x)}{x^2+a^2}\,dx=\frac {\pi e^{-a}}{2a} \] where \(a>0\).