Lebesgue Integration

Let

Show that \(A\) has Lebesgue measure 0 but

Let \(f:\mathbb {R}\to \mathbb {C}\) be Lebesgue measurable. Assume that \(f\) is invariant under translations by rational numbers. Show that \(f\) is constant Lebesgue almost everywhere.

Let \(f\) be Lebesgue integrable on \((0,1)\). For \(x\in (0,1)\) define \[ g(x)=\int _x^1 \frac {f(t)}{t}\,d\lambda . \]

Prove that \(g\) is Lebesgue integrable on \((0,1)\) and that \[ \int _0^1 g(x)\,d\lambda =\int _0^1 f(x)\,d\lambda . \]

Let \((X,\mathcal {B},\mu )\) be a finite measure space and \(f:X\to \mathbb {R}\) be measurable.

1. Show that if \(g_n(x)=(f(x))^n\) and \(\int g_n\,d\mu \) is uniformly bounded for all \(n\) then \(|f(x)|\le 1\) (\(\mu \)-a.e.).
2. Show that \(\int _X g_n\,d\mu \) is independent of \(n\) iff there is \(A\in \mathcal {B}\) so that \(f=\chi _A\) (\(\mu \) a.e.).

Let \((X,\mathcal {M},\mu )\) and \((Y,\mathcal {N},\nu )\) be two measure spaces and \(A\) be measurable with respect to \(\mathcal {M}\times \mathcal {N}\), the \(\sigma \)-algebra generated by measurable rectangles. Show that if \(A_1\subset A_2\subset \cdots \) satisfy that \[ (\mu \otimes \nu )(A_i)=\int _X \nu (\{y:(x,y)\in A_i\})\,d\mu (x) \]

for all \(i\) then

In this problem, let \(\lambda \) denote Lebesgue measure on \(\mathbb {R}\) and \(\widehat {\cdot }\) denote the Fourier transform (on \(\mathbb {R}\)). Show that if \(f,g\) are Lebesgue integrable and \(\hat f=\hat g\) then \(f=g\) \(\lambda \)-a.e.

Let \(f : [0, 1] \to [0, \infty ]\) be measurable. Show that \(U(f) = \{(x, y) : 0 \le y \le f(x)\}\) is measurable and \[ \int f\,d\lambda = \lambda ^{2}(U(f)). \]

State the Lebesgue dominated convergence theorem. Show that the converse is not true. That is, there is a sequence of functions that satisfy the conclusion of the Lebesgue dominated convergence theorem but not its assumption.

1. Let \((X, \mathcal {M})\) be a measurable space, \(\mu , \nu \) positive measures on \((X, \mathcal {M})\). Suppose \(\nu (X) < \infty \). Prove that \(\nu \) \(\ll \mu \) if and only if for every \(\epsilon > 0\) there exists \(\delta > 0\) such that \(\mu (E) < \delta \) implies \(\nu (E) < \epsilon \).
2. For \(\mu \) the Lebesgue measure on \(\mathbb {R}\) find a positive (but infinite) measure \(\nu \) such that \(\nu \) \(\ll \mu \) but the \(\epsilon \)-\(\delta \) statement in (a) fails.

Show that there does not exist a Lebesgue measurable subset \(A \subset \mathbb {R}\) such that for every interval \((a, b) \subset \mathbb {R}\) we have \[ m(A \cap (a, b)) = \frac {b-a}{2} \] where \(m\) denotes Lebesgue measure.

Let \(f : [0, 1] \to \mathbb {R}\) be a continuous function and define a signed measure \(\mu \) on the Lebesgue measurable subsets of \([0, 1]\) by

where \(m\) is Lebesgue measure. Suppose that for all \(n = 0, 1, 2, 3, \ldots \)

Prove that \(\mu = 0\).

Let \(f, g \in L^{2}(\mathbb {R})\). Show that

is a well-defined real number for every \(x \in \mathbb {R}\) and that \(h : \mathbb {R} \to \mathbb {R}\) is continuous.

1. State the Monotone Convergence Theorem.
2. Let \(f : [0, \infty ) \to \mathbb {R}\) defined by \(f(x) = e^{-x}\). Explain why \(f\) is Lebesgue integrable and compute its integral. You may use the fact that if a function \(f : [0, \infty ) \to \mathbb {R}\) is 0 outside an interval \([a, b]\) and \(f|_{[a,b]} : [a, b] \to \mathbb {R}\) is Riemann integrable then \(f\) is Lebesgue integrable and the Lebesgue integral of \(f\) equals the Riemann integral of \(f|_{[a,b]}\).

Let a measure space \((X, \mathcal {M}, \mu )\) be given and for a fixed measurable function \(f : X \to [0, \infty )\) define a new measure \(\lambda \) on \(\mathcal {M}\) by \[ \lambda (E) = \int _{E} f\,d\mu . \]

You don’t have to prove that \(\lambda \) is a measure. Prove that for any measurable \(g : X \to [0, \infty )\), \[ \int g\,d\lambda = \int gf\,d\mu . \]

Let \((X,\mathcal M,\mu )\) be a measure space and let \(f:X\to [0,\infty )\) be a measurable function. Define a new measure \(\nu \) by \[ \nu (E)=\int _E f\,d\mu \] for \(E\in \mathcal M\) (you don’t have to prove that \(\nu \) is a measure). Prove that for every positive measurable function \(g\), \[ \int _X g\,d\nu =\int _X f g\,d\mu . \]

Let \((X,\mathcal M,\mu )\) be a measure space with \(\mu (X)<\infty \). Denote by \(S\) the set of equivalence classes of measurable functions \(X\to \mathbb R\) where \(f\sim g\) if and only if \(f=g\) a.e. For \(f,g\in S\) define \[ d(f,g)=\int _X\frac {|f(x)-g(x)|}{1+|f(x)-g(x)|}\,d\mu . \] Show that \((S,d)\) is a metric space and \(f_k\to f\) in \((S,d)\) if and only if \(f_k\to f\) in measure in \(X\).

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

Let \(f:[0,\infty )\to \mathbb R\) be defined by \(f(x)=e^{-x}\). Explain why \(f\) is Lebesgue integrable and compute its integral.

Using the methods of complex analysis compute \[ \int _0^\infty \frac {1}{x^4+16}\,dx. \]

Let \(f\ge 0\) be an integrable function on a measure space \(X\). Prove that, for every \(\epsilon >0\), there exists a set \(E\subset X\) of finite measure such that \[ \int _E f>\left (\int _X f\right )-\epsilon . \]

Compute \[ \int _0^{2\pi }\frac {d\theta }{4+\cos (\theta )} \] using the methods of complex analysis.

Using the methods of complex analysis, compute \[ \int _0^\infty \frac {x^2}{x^4+1}\,dx \] and simplify your answer.

Let \((X,\mu )\) be a (not necessarily finite) measure space and \(f\ge 0\) be a \(\mu \) integrable function. Prove that, for every \(\epsilon >0\), there exists a set \(E\subset X\) of finite measure such that \[ \int _E f\,d\mu >\int _X f\,d\mu -\epsilon . \]

Let \(f:[0,\infty )\to \mathbb R\) be defined by \(f(x)=e^{-x}\). Explain why \(f\) is Lebesgue integrable and compute its integral.

Use the methods of complex analysis to compute \[ \int _{-\infty }^{\infty }\frac {2}{(x^2+4)(x^2+1)}\,dx. \]

Let \(f:[0,\infty )\to \mathbb R\) be defined by \(f(x)=e^{-x}\). Explain why \(f\) is Lebesgue integrable and compute its integral.

Calculate \[ \int _\gamma (1+z^2)\sin (1/z)\,dz \] where \(\gamma \) is the unit circle traversed in the counter-clockwise direction.

Show that the equation \[ e^z=2z+1 \] has exactly one solution in \(D\).

Hint: Note that \[ e^z-1=\int _0^1 e^{tz}z\,dt. \] Conclude that \(|e^z-1|\le e-1\).

Let \(f\) be a function holomorphic for \(z\) in the annulus \(r<z<R\). Show that if for some \(\rho \) with \(r<\rho <R\), \[ \int _{|z|=\rho } z^n f(z)\,dz=0 \] for all nonnegative \(n\), then \(f\) extends to a holomorphic function on the disk \(|z|<R\).

Using methods of complex analysis, evaluate \[ \int _0^\infty \frac {x^2}{x^4+x^2+1}\,dx. \]

Show that if \(f:[0,\infty )\to \mathbb C\) is continuous and compactly supported, then \[ \lim _{\rho \to 1}\int |f(x^\rho )-f(x)|\,d\lambda =0. \] Use this to show that if \(f\in L^\infty ([1,2],\lambda )\), we have that \[ \lim _{\rho \to 1^+}\int |f(x^\rho )-f(x)|\,d\lambda =0. \]

Calculate \[ \int _\gamma (1+z^2)e^{1/z}\,dz \] where \(\gamma \) is the unit circle traversed once in the counter-clockwise direction.

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

If \(f\in L^1(\lambda )\) then \[ \lim _{c\to 1}\int |f(cx)-f(x)|\,d\lambda =0. \]

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

A family of complex-valued functions \(\mathcal S\) on \([0,1]\) is uniformly integrable if for all \(\epsilon >0\) there exists \(\delta >0\) so that \(\lambda (A)<\delta \) implies \(\left |\int _A f\,d\lambda \right |<\epsilon \) for all \(f\in \mathcal S\).

Let \((f_i;i\in \mathbb N)\) be a uniformly integrable sequence. Assume that \[ \lim _{i\to \infty }f_i(x)=g(x)\quad \text {for all }x\in [0,1]. \] Prove that \[ \lim _{i\to \infty }\int _0^1 f_i\,d\lambda =\int _0^1 g\,d\lambda . \]

Let \(f:\mathbb R^2\to \mathbb C\) be a Borel measurable function. Let \(g:\mathbb R^2\to \mathbb R^2\) be the map given by \(g(x,y)=(2x+y^2,y)\) for \((x,y)\in \mathbb R^2\). If \(f\in L^1(\lambda ^2)\), what is \[ \frac {\int _{\mathbb R^2}(f\circ g)\,d\lambda ^2} {\int _{\mathbb R^2}f\,d\lambda ^2}? \]

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

(a) If \(f_i\to g\) are in \(L^1(\mathbb R,\lambda )\) and \(\phi \in C(\mathbb R)\), does \[ \lim _{i\to \infty }\int f_i\phi \,d\lambda =\int g\phi \,d\lambda ? \]

(b) Show that if \(f_i\to g\) in \(L^1(\lambda ,[0,1])\), then there exists a subsequence \(n_1,\ldots \) so that \(f_{n_i}\to g\) almost everywhere.

Let \(f\) be in \(L^1(\lambda ,\mathbb R)\). Show that for any \(r>0\) the function \[ h(s)=\int _{s-r}^{s+r} f\,d\lambda \] is continuous.

What is \[ \int _{-\infty }^{\infty }\frac {(\cos x)^2}{x^2+5}\,dx? \]

Show that if \(f,g\in L^1(\lambda ,\mathbb R)\), then \(\lVert f*g\rVert _1\le \lVert f\rVert _1\lVert g\rVert _1\). Recall \[ (f*g)(x)=\int f(t)g(x-t)\,d\lambda (t). \]

Evaluate \[ \int _C \frac {1}{z^2-z}\,dz \] where \(C\) is a closed path in \(\mathbb C-[0,1]\).

Let \(f_i:[0,1]\to \mathbb R\) be a sequence of functions that converge pointwise to \(f_\infty \).

(a) Show that if the \(f_i\) are uniformly bounded then \[ \lim _{i\to \infty }\int f_i\,d\lambda =\int f_\infty \,d\lambda . \]

(b) What if \(\lVert f_i\rVert _1\le 1\) for all \(i\)?

(c) What if \(\lVert f_i\rVert _2\le 1\) for all \(i\)?

Let \(f\in L^1(\lambda ,[0,1])\) have the property that for any measurable set \(A\) with \(\lambda (A)=\frac {1}{\pi }\) we have \(\int _A f\,d\lambda =0\). Show that \(f=0\) almost everywhere.

Let \(f\) be an a holomorphic function defined on an open set \(U\subset \mathbb C\) and \(a\in U\). Assume that \(f'(a)\) is nonzero. Show that for sufficiently small circles \(C\) centered at \(a\) \[ \frac {2\pi i}{f'(a)}=\int _C \frac {dz}{f(z)-f(a)} \]

Evaluate \[ \int _C \frac {1}{(z+i)(z-1)}\,dz \] where \(C\) is pictured on the next page:

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Let \((X,M,\mu )\) be a measure space, and suppose that \(f:X\times [0,1]\to \mathbb R\) such that \(f(\cdot ,t):X\to \mathbb R\) is integrable for all \(t\). Define \[ F(t)=\int _X f(x,t)\,d\mu (x) \] for \(t\in [0,1]\). Suppose that \(\frac {\partial f}{\partial t}(x,t)\) exists for all \(x\) and \(t\), and that for some integrable function \(g\), \(\left |\frac {\partial f}{\partial t}(x,t)\right |\le g(x)\) for all \(x\) and \(t\). Show that \(F\) is differentiable and that \[ F'(t)=\int _X \frac {\partial f}{\partial t}(x,t)\,d\mu (x) \]

Evaluate \[ \int _\Gamma \frac {\sin z}{z^6}\,dz \] where \(\Gamma \) is a positively oriented simple unit circle \(\{z\in \mathbb C:|z|=1\}\).

Let \(f\in L^p(\mathbb R)\) (with respect to the Lebesgue measure), and \(1\le p\le \infty \), and let \[ F(x)=\int _0^x f(t)\,dt. \] Prove that \(F\) defines a continuous function on \(\mathbb R\).

Let \((X,\mathcal M,\mu )\) be a measure space, and let \(f\ge 0\) be an integrable function on \(X\). Suppose that \(f_n\ge 0\) is a measurable function, \(n=1,2,\ldots \), and that \[ f_1\le f_2\le \cdots \le f. \] If \(\int f_n\,d\mu \to \int f\,d\mu \), prove that \(f_n\to f\) a.e.

Let \((X,\mathcal M,\mu )\) be a measure space, \(1<p<\infty \), and \(q=p/(p-1)\). Suppose that \(f_n\in L^p(X)\), \(n=1,2,\ldots \), and that for each \(g\in L^q(X)\), \[ \sup _n \left |\int f_ng\,d\mu \right |<\infty . \] Show that \(\sup _n\lVert f_n\rVert _p<\infty \).

Consider \(\mathbb R\) with the Lebesgue measure, and recall the definitions of convolution and the Fourier transform, for \(f,g\in L^1(\mathbb R)\): \[ f*g(x)=\int f(x-y)g(y)\,dy,\qquad \hat f(\xi )=\int f(x)e^{-2\pi ix\xi }\,dx. \] Show that for \(f,g\in L^1(\mathbb R)\), \(f*g\in L^1(\mathbb R)\), and that \[ \widehat {f*g}=\hat f\,\hat g. \]

Evaluate the integral \[ \int _0^\infty \frac {x^2+1}{x^4+1}\,dx. \]

Let \(f\) be a continuous function on the circle \(\{e^{i\theta }\mid 0\le \theta <2\pi \}\). Let \[ c_n:=\frac {1}{2\pi }\int _0^{2\pi } f(e^{i\theta })e^{-in\theta }\,d\theta . \] Prove or disprove: \[ f(e^{i\theta })=\lim _{N\to \infty }\left (\sum _{n=-N}^{N}c_ne^{in\theta }\right ) \] for all \(\theta \).

Let \((X,\mu )\) be a measure space, let \(\{f_n\}\) be a sequence of nonnegative integrable functions, and assume there exists an integrable function \(f\) such that \(f(x)=\lim _{n\to \infty } f_n(x)\) almost everywhere. Further assume \[ \tag {1} \lim _{n\to \infty }\int _X f_n\,d\mu =\int _X f\,d\mu . \] Prove that \[ \lim _{n\to \infty }\int _X |f_n-f|\,d\mu =0. \] Show by example that this need not hold if the assumption (1) is omitted.

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

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

Let \(n\) be a positive integer. Use a contour integral to compute \[ \int _0^{2\pi }\cos ^{2n}(\theta )\,d\theta \] in terms of \(n\).

Recall that the Fourier transform of a function \(f\in L^1(\mathbb R)\) (with respect to Lebesgue measure) is given by \[ \hat f(x)=\int _{-\infty }^{\infty } e^{-2\pi ixy}f(y)\,dy \] Show that \(\hat f\) is a bounded, continuous function on \(\mathbb R\).

Calculate \[ \int _\gamma (1+z^2)e^{1/z}\,dz \] where \(\gamma \) is the unit circle traversed in the counter-clockwise direction.

Let \(f\) be a function holomorphic for \(z\) in the annulus \(r<z<R\). Show that if for some \(\rho \) with \(r<\rho <R\), \[ \int _{|z|=\rho } z^n f(z)\,dz=0 \] for all integers \(n\) with \(n\ge 0\) then \(f\) extends to a holomorphic function on the disk \(|z|<R\).

Decide whether the following statement is true or false. If true, prove it, if false, provide a counterexample.

Let \(\{f_n\}\) be a sequence of measurable functions (with respect to Lebesgue measure) with \(0\le f_n\le 1\) such that \(\int _0^1 f_n(x)\,dx\to 0\). Then \(f_n\to 0\) almost everywhere.

Let \(f\in L^1(\mathbb R)\) such that \[ \int _{-\infty }^{\infty }|x\hat f(x)|\,dx<\infty \] Prove that there is a differentiable function \(g\) on \(\mathbb R\) such that \(f=g\) almost everywhere.

Let \(f\) be a holomorphic function on \(\mathbb C\setminus \{0\}\) such that \(f(z)\sin z\to 1\) as \(z\to 0\). Calculate \[ \int _\gamma f(z)\,dz \] where \(\gamma \) is the unit circle traversed counter-clockwise.

Let \(f:\Omega \to \mathbb C\) be a complex-valued function on an open domain \(\Omega \subset \mathbb C\). Assume that for all closed contours \(\gamma \) that consists of line segments parallel to the real or imaginary axes \[ \int _\gamma f(z)\,dz=0. \] Show that there exists a holomorphic function \(F:\Omega \to \mathbb C\) such that \(F'(z)=f(z)\).

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 \(z=e^{i\theta }\).

Let \(f:[0,1]\to \mathbb R_{\ge 0}\) be a measurable integrable function \(f\in L^1([0,1])\) (with respect to the Lebesgue measure). Show that \[ \int _{[0,1]} x f(x)\,dx =\int _{[0,1]}\left (\int _{[y,1]} f(x)\,dx\right )\,dy \]

Compute the integral \[ \int _0^\infty \left (\frac {\sin x}{x}\right )^2\,dx. \]

Let \((X,\mu )\) be a measure space and \(f\ge 0\) an integrable function on \(X\). Prove that for \(\epsilon >0\), there exists \(\delta >0\) such that whenever \(\mu (A)<\delta \), then \(\int _A f<\epsilon \).

Let \(K\) be a continuous function on the square \([0,1]\times [0,1]\), and let \([0,1]\) be endowed with the Lebesgue measure. For \(f\in L^2([0,1])\) define \[ Tf(x)=\int _0^1 K(x,y)f(y)\,dy. \] Show that \(Tf\in L^2([0,1])\) and that \(T:L^2\to L^2\) is a bounded operator with \(\lVert T\rVert \le \lVert K\rVert _\infty \).

Let \(f\in L^1([-\pi ,\pi ])\), and let \(c_n\) be the nth Fourier coefficient of \(f\): \[ c_n=\frac {1}{\sqrt {2\pi }}\int _{-\pi }^{\pi } f(x)e^{-inx}\,dx \quad n\in \mathbb Z \] Prove that \(c_n\to 0\) as \(|n|\to \infty \).

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 \(f\) be integrable on \(\mathbb R\) with respect to Lebesgue measure, and define \[ F(x)=\int _{-\infty }^{x} f(t)\,dt \] Prove that \(F\) is continuous.

Let \(f\in L^1([-\pi ,\pi ])\) and for \(n\in \mathbb Z\) let \(c_n\) be the nth Fourier coefficient of \(f\), that is \[ c_n=\frac {1}{\sqrt {2\pi }}\int _{-\pi }^{\pi }f(x)e^{-inx}\,dx \] Prove that \(c_n\to 0\) as \(|n|\to \infty \).

Let \(I=[0,1]\) be the unit interval with the Lebesgue measure, and let \(I\times I\) be endowed with the product measure. Assume that \(K\in L^2(I\times I)\) and \(f\in L^2(I)\). Define \[ g(x)=\int _0^1 K(x,y)f(y)\,dy \] Show that \(g\in L^2(I)\).

Calculate the integral \[ \int _{-\infty }^{\infty }\frac {\cos x}{x^2+4}\,dx. \]

Assume that \(f\) is non-negative and \[ \int _X f\,d\mu <\infty . \] Show that for every \(\epsilon >0\) there exists a \(\delta >0\) such that if \(A\) is a measurable subset of \(X\) such that \(\mu (A)\le \delta \) then \[ \int _A f\,d\mu \le \epsilon . \]

Let \(E_1,\ldots ,E_n\) by a finite collection of disjoint measurable subsets of \(X\) and let \(M\) be a subspace of \(L^2(X)\) spanned by the functions \(\{u_1,\ldots ,u_n\}\) where each \(u_i\) is in \(L^2(X)\) and has support contained in \(E_i\). Let \(P:L^2(X)\to \mathbb R\) be the orthogonal projection. Find a function \(h:X\times X\to \mathbb R\) such that \[ Pf(x)=\int _X h(x,y)f(y)\,d\mu (y) \] for all \(f\in L^2(X)\).

Let \(f\) be a holomorphic function on an open set containing \(\overline {D_1}(0)\). Prove that if \(z\) is in \(D_1(0)\), then \[ f(z)=\frac {1}{2\pi i}\int _\gamma \frac {f(w)}{\sin (w-z)}\,dw \] where \(z\in D_1(0)\) and \(\gamma \) traverses the unit circle once in the positive direction.

Evaluate the integral \[ \int _0^{2\pi }\frac {\cos ^2 3\phi }{1-2a\cos \phi +a^2}\,d\phi \] where \(a\) is a complex number such that \(|a|<1\).

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 \(f\).

Construct a sequence of continuous real-valued functions \(f_n\) on \([0,1]\) such that \(0\le f_n\le 1\) and \[ \lim _{n\to \infty }\int _0^1 f_n(x)\,dx=0, \] but such that the sequence \(\{f_n(x)\}\) converges for no \(x\in [0,1]\).

Provide a rigorous computation of the sum of the following series: \[ \sum _{n=1}^{\infty }\frac {1}{n^4}. \] (Hint: one possible approach is to consider the integral of \(f(z)=\frac {\pi }{z^4}\cot (\pi z)\) over an appropriate sequence of contours.)

Determine the region of values of \(z\) for which \[ f(z)=\int _0^\infty \frac {e^{tz}}{1+t^2}\,dt \] defines a holomorphic function. (Hint: one approach is to use the holomorphicity criterion of Morera’s Theorem. If you pursue this approach, make certain to justify any application of Fubini’s theorem.)

Let \(\{f_n\}\) be a sequence of complex valued measurable functions defined on a finite interval \((a,b)\subset \mathbb R\) (”measurable” means with respect to the Lebesgue measure). Suppose that there exists a constant \(M<\infty \) such that \(\int _a^b |f_n(x)|^2\,dx<M\) for all \(n\), and \(f_n(x)\to 0\) a. e. Prove that \[ \lim _{n\to \infty }\int _a^b f_n(x)\,dx=0. \] Show also that this assertion fails in general if one replaces the interval \((a,b)\) by \(\mathbb R\) (give a counterexample).

Let \(\Omega \) open in \(\mathbb C\), \(b\in \Omega \), \(f\in H(\Omega )\), and assume that \(f'(b)\ne 0\). Show that \[ \frac {2\pi i}{f'(b)}=\int _C \frac {1}{f(z)-f(b)}\,dz \] for sufficiently small positively oriented simple circles centered at \(b\).

Let \(f_1,f_2,\cdots ,f_n,\cdots \) be a sequence of functions in \(L^1(\mathbb R)\) such that \[ \lim _{n\to \infty }\int _{\mathbb R} f_n(x)g(x)\,dx=0 \] for every \(g\in L^\infty (\mathbb R)\). Show that \(f_1,f_2,\cdots ,f_n,\cdots \) is a bounded sequence in \(L^1(\mathbb R)\).

Evaluate the integral \[ \int _0^\infty \frac {\sin ax}{x(x^2+b^2)}\,dx \] where \(a\) and \(b\) are real numbers and \(b\ne 0\).

Let \(\mu \) be a positive measure on a measure space \((X,\Sigma )\) and let \(f\) be a complex valued function on \(X\) which is integrable with respect to \(\mu \). For each \(n\), set \[ E_n=\{x\in X:1/n\le |f(x)|\le n\}. \] Then each \(E_n\) is a measurable set on which \(|f|\) is bounded. Prove that

(a) \(\mu (E_n)\) is finite for each \(n\); and

(b) \(\displaystyle \lim _{n\to \infty }\int _{E_n} f(x)\,d\mu (x) =\int _X f(x)\,d\mu (x)\).

Consider \(L^1(T)\) to be the space of periodic functions on \(\mathbb R\) of period \(2\pi \) which are integrable on \([0,2\pi ]\). Let \(M\) be a proper closed translation invariant subspace of \(L^1(T)\).

(a) Prove that there exists a non-zero \(h\in L^\infty (T)\) such that \(f*h(x)=0\) for all \(f\in M\), \[ \text {where } f*h(x)=\frac {1}{2\pi }\int _0^{2\pi } f(x-y)h(y)\,dy; \]

(b) use part (a) to prove there must be an \(n\in \mathbb Z\) such that \(\hat f(n)=0\) for every \(f\in M\), where \(\hat f(n)\) is the \(n\) th Fourier coefficient of \(f\).

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 \(z=e^{i\theta }\).

Let \(\alpha (x,y)\) be a continuous function on \([0,1]\times [0,1]\). Show that \[ A(f)(x)=\int _0^1 \alpha (x,y)f(y)\,dy \] defines a bounded operator from \(L^2([0,1])\) to \(C([0,1])\), considered as a Banach space with respect to \(\|f\|=\sup _x |f(x)|\).

Evaluate the integral: \[ \int _0^\infty \frac {dx}{1+x^3} \]

Prove that if \(f,g\in L^2(\mathbb R)\) and the convolution product \(f*g\) is defined by \[ f*g(x)=\int f(y)g(x-y)\,dy, \] then \(f*g(x)\) is a continuous function which vanishes at infinity.

Let \(H_n\) be the linear span in \(L^2([-\pi ,\pi ])\) of the set \(\{e^{ikt}\}_{k=-n}^n\). The orthogonal projection \(P_n\) of \(L^2([-\pi ,\pi ])\) onto \(H_n\) may be written in the form \[ P_n f(x)=h_n*f(x)=\frac {1}{2\pi }\int _{-\pi }^{\pi }h_n(x-t)f(t)\,dt. \] Find the function \(h_n\) and prove that it has this property.

Prove that

(a) \(\log (1+e^t)\le t+\log (2)\) for all \(t\ge 0\);

(b) if \(f\) is a real valued function in \(L^1([0,1])\) then \[ \lim _{n\to \infty }\frac {1}{n}\int _0^1 \log (1+e^{n f(x)})\,dx =\int _0^1 f^+(x)\,dx, \] where \(f^+(x)=f(x)\) if \(f(x)\ge 0\) and \(f^+(x)=0\) if \(f(x)<0\).

Suppose \(f\) is holomorphic in \(r<|z|<R\), and suppose that for some \(\rho \), \(r<\rho <R\), \[ \int _{|z|=\rho } f(z)z^n\,dz=0 \] for all negative integers \(n\). Show that \(f\) has a holomorphic extension to the set \(|z|>r\).

Let \(\gamma \) be a closed path in \(\mathbb C\setminus [0,1]\). Show that \[ \int _\gamma \frac {1}{z(1-z)}\,dz=0. \]

Let \(f\) be a non-negative measurable function and \(\mu \) a finite positive measure on a measure space \((X,\Sigma )\). Prove that \[ \lim _{n\to \infty }\int \frac {f^n(x)}{1+f^n(x)}\,d\mu (x) =\mu (E)+\frac {1}{2}\mu (F), \] where \(E=\{x\in X; f(x)>1\}\) and \(F=\{x\in X; f(x)=1\}\).

Let \(\mu \) be a measure on a measure space \(X\) and let \(f\ge 0\) be a measurable function on \(X\). Prove that \[ \int _X f(x)\,d\mu (x)=\int _0^\infty \phi (y)\,dy \] where \(\phi (y)=\mu (\{x:f(x)\ge y\})\).

Compute \[ \int _0^\infty \frac {x^2}{x^4+1}\,dx. \]

Let \(f_n\) be a sequence of functions in \(L^\infty (\mathbb R)\) such that \[ \sup _n\left |\int _{\mathbb R} f(x)g(x)\,dx\right |<\infty \] for all \(g\in L^1(\mathbb R)\). Show that \(\sup _n\lVert f_n\rVert <\infty \).

Using methods of complex analysis, evaluate \[ \int _0^\infty \frac {x^2\,dx}{x^4+x^2+1} \]

Let \(f\in L^1(\mathbb R)\). Define a function \(\varphi \) on \(\mathbb R\) by the formula \(\varphi (x)=\int _{-\infty }^{x}f(t)\,dt\). Show that \(\varphi \) is a uniformly continuous function on \(\mathbb R\).

Evaluate \[ \int _{\Gamma }\frac {\cos z}{z^5}\,dz \] where \(\Gamma \) is a positively oriented simple unit circle \(\{z:|z|=1\}\).

Recall the \(k\)th Fourier coefficient of \(f\in L^1([-\pi ,\pi ])\) is \[ \hat f(k)=\frac {1}{2\pi }\int _{-\pi }^{\pi }f(t)e^{-ikt}\,dt. \]

(1) If \(f(t)=t\) for \(-\pi \le t<\pi \), show that \(\hat f(k)=(-1)^{k+1}\frac {1}{ik}\) for \(k\ne 0\) and \(\hat f(0)=0\);

(2) Show that \(\sum _{k=1}^{\infty }k^{-2}=\frac {\pi ^2}{6}\).

Find \[ \int _{\gamma _R}\frac {dz}{e^z-1} \] where \(\gamma _R\) is the curve \(|z|=R\) traversed counterclockwise and \(R\) is a positive number such that \(\gamma _R\) does not pass through a zero of \(e^z-1\).

Let \(\mathcal F=\{f_\alpha \}_{\alpha \in A}\) be a family of continuous complex valued functions on a compact Hausdorff space. Let \(\{c_\alpha \}_{\alpha \in A}\) be a family of complex numbers. Suppose that for each \(F\subset A\) with \(F\) finite, there exists a complex measure \(\mu _F\) of total variation one on \(X\) such that \[ \int _X f_\alpha \,d\mu _F=c_\alpha \qquad (\forall \alpha \in F). \] Show that there exists a complex measure \(\mu \) on \(X\) of total variation at most one such that \[ \int _X f_\alpha \,d\mu =c_\alpha \qquad (\forall \alpha \in A). \]

Suppose that \((X,\mathfrak M,\mu )\) is a measure space with \(\mu \) a finite positive measure. Let \(I\) be an open interval in \(\mathbb R\) and \(f:X\times I\to \mathbb C\) a bounded function such that \(x\to f(x,t)\) is \(\mathfrak M\)-measurable for each \(t\in I\) and \(t\to f(x,t)\) is differentiable for each \(x\in X\). If \(\frac {d}{dt}f(x,t)\) is also bounded on \(X\times I\) prove that \(h(t)=\int _X f(x,t)\,d\mu (x)\) is differentiable on \(I\) and in fact \[ h'(t)=\int _X \frac {d}{dt}f(x,t)\,d\mu (x). \]

Let \(f\) be holomorphic in an open set containing the disk \(D=\{z\in \mathbb C:|z|\le r\}\) and suppose that the only zero of \(f\) in \(D\) is a simple zero at \(a\) belonging to the interior of \(D\). If \(C_r\) is the positively oriented contour \(|z|=r\) show that \[ \int _{C_r}\frac {1}{f(z)}\,dz=\frac {2\pi i}{f'(a)}. \]

Use methods of complex analysis to find \[ \int _{-\infty }^{\infty }\frac {dx}{x^4+1}. \]

Let \(H=L^2(\mathbb R)\) be the Hilbert space with the inner product \(\int _{\mathbb R}f(x)\bar g(x)\,dx\), where \(dx\) is the Lebesgue measure. Let \(T:H\to M\) be the orthogonal projection of \(H\) onto a finite dimensional subspace \(M\) of \(H\).

Show that there exists a function \(h:\mathbb R^2\to \mathbb C\) such that \[ Tf(x)=\int _{\mathbb R}h(x,y)f(y)\,dy \] for all \(f\in L^2(\mathbb R)\).

(Hint: Let \(u_1,\ldots ,u_n\) be an orthonormal basis of \(M\). You may express \(h\) in terms of \(u_1,\ldots ,u_n\).)

Let \(X\) be as in Problem 2. Give an example of a sequence of measurable, nonnegative functions \(f_n:X\to \mathbb R\), \(f_1\ge f_2\ge \cdots \ge f_n\ge \cdots \), converging pointwise to zero, such that \(\int _X f_n(x)\,dx\) does not converge to zero.

Can any of the integrals \(\int _X f_n(x)\,dx\) be finite? Justify your answer.

Evaluate \[ \int _{\Gamma }\frac {e^z}{z^3}\,dz \] where \(\Gamma \) is a positively oriented simple unit circle \(\{z:|z|=1\}\).

Let \((X,\mathcal M,\mu )\) be a measure space, where the measure \(\mu \) is positive. Suppose \(f,f_n\in L^1(X,\mu )\), \(f_n\ge 0\) for each \(n\), \(f_n\to f\) a.e. and \[ \int _X f_n\,d\mu \to \int _X f\,d\mu \] Prove that \(f_n\to f\) in \(L^1(X,\mu )\) norm.

Hint: Show first that \((f-f_n)^+\le f\) for each \(n\) (recall that for real valued function \(g\), \(g^+=\max (g,0)\)).

In \(L^2((0,2\pi ))\) consider the orthogonal projection onto the subspace spanned by \(\{e^{-it},1,e^{it}\}\). Express this projection as an integral operator (i.e. an operator of the form \(f\to Pf\), where \(Pf(y)=\int _0^{2\pi }f(x)h(x,y)\,dx\)).

Find \[ \int _{|z|=5}\tan (z)\,dz. \]