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• Bonds and CDs. Updated 3/6/2017, 3/3/2017, 2/21/2017, 2/20/2017.
  A brief introduction is given to compound interest, certificates of deposit, and bonds. The focus is on determining a fair price, yield to maturity, accrued interest, and duration. Matlab code is given to compute the accrued interest with the 30/360 US method, which is used for US corporate bonds and many US agency bonds.
  Keywords: banker's rule, certificate of deposit, coupon, days between dates, discount, exact interest, Macaulay duration, modified duration, ordinary interest, premium, present value, sensitivity.

• Introduction to Mathematical Induction. Updated 2/13/2022.

• Magnitude and Phase of Complex Numbers. Updated 3/30/2021, 7/12/2017, 7/6/2017.
  Every nonzero complex number can be expressed in terms of its magnitude and angle. This angle is sometimes called the phase or argument of the complex number. Although formulas for the angle of a complex number are a bit complicated, the angle has some properties that are simple to describe. In particular, when the complex number is a function of frequency, we derive a simple formula for the derivative of the argument.

• Why Are Minimum-Phase Systems Called Minimum Phase? Updated 7/12/2017, 7/9/2017.
  The main reason is that such systems have minimum phase lag (phase lag is the negative of the phase). The key to deriving this result is to study the derivative of the phase, which is closely related to the group delay. This further requires a deeper-than-usual understanding of the angle or argument of a complex number. This background is included.

• Gaussian Quadrature and the Eigenvalue Problem. (Printing options do not apply.)
  Introduction to Gaussian quadrature (numerical integration) and orthogonal polynomials.
  Keywords: Boole's rule, Newton–Cotes formulas, Simpson's rule, three-term recurrence.

• Introduction to Frequency Analysis and the DFT (Short Version). Updated 6/1/2017, 5/25/2017.
  The continuous-time Fourier transform is defined. It is shown that windowing has the effect of blurring the transform. Then sampling is introduced as a way to approximate the transform integral. The resulting approximation is seen to be periodic. Windowing and sampling are then combined to obtain the discrete Fourier transform (DFT). Matlab code for plotting the fast Fourier transform (FFT) on the appropriate frequency axis is provided.
  Keywords: Nyquist rate, sampling rate, sinc function, spectrum.

• Introduction to Frequency Analysis and the DFT (Long Version). Updated 5/25/2017, 4/23/2016.
  The relationships between the continuous-time Fourier transform, the continous-time Fourier series, and the discrete Fourier transform (DFT) are presented.
  Keywords: fast Fourier transform (FFT), line spectrum, Nyquist rate, sampling rate, sinc function, spectrum.

• Derivation of the Fourier Inversion Formula, Bochner's Theorem, and Herglotz's Theorem. Updated 2/1/2024 (on p. 3, added eq # for inv. Fourier transform formula), 6/14/2023 (updated text at end of proof of Bochner's Theorem), 6/13/2023 (added eq. number to proof of Lemma 8), 6/6/2023 (changed sentence on p. 8), 6/4/2023 (p. 7, corrected formula for δ_n(t) --- thank you reader!, and B_ν), 6/1/2023 (corrections to the Example after Theorem 4 --- thank you reader!), 4/8/2018.
  Summability kernels are introduced in order to rigorously approximate the Dirac delta function and its sifting property. It is then possible to establish the convolution property of the Fouier transform and the inverse Fourier transform formula. The inversion formula is then used to motivate Bochner's theorem. Bochner's theorem is proved by appealing to well-known properties of limits of sequences of cumulative distributions in probability theory. Herglotz's theorem is handled similarly.
  Keywords: Pareseval's equation.

• The Intermediate-Value Theorem. Updated 12/20/2020, 2/15/2015, 11/9/2014.
  A simple proof of the intermediate-value theorem is given. As an easy corollary, we establish the existence of $n$th roots of positive numbers.

• Taylor's Theorem with Remainder and Stirling's Formula. Updated 6/25/2021, 6/24/2021, 6/23/2021.
  Taylor's Theorem with Remainder is stated and applied to some simple examples. A proof of the theorm is given. The theorem is then used to derive Stirling's Formula.

• The Gamma Function and Stirling's Formula. Updated 7/2/2021, 6/23/2021.
  Starting with Euler's integral definition of the gamma function, we state and prove the Bohr-Mollerup Theorem, which gives Euler's limit formula for the gamma function. We then discuss two independent topics. The first is upper and lower bounds on the gamma function, which lead to Stirling's Formula. The second is the Euler-Mascheroni Constant and the digamma function.
  Keywords: integral definition, limit formula.

• Basic Properties of Power Series. Updated 11/9/2014, 10/4/2014, 10/3/2014, 10/2/2014, 9/30/2014, 9/23/2014, 9/21/2014.
  These notes provide a quick introduction (with proofs) to the basic properties of power series, including the exponential function and the fact that power series can be differentiated term by term.
  Keywords: absolute convergence, comparison test, discrete Fubini theorem, differentiation term by term, exponential function, geometric series, multiplication of power series, radius of convergence, uniform Cauchy criterion, uniform convergence.

• The Bezout Identity and an Application. Updated 5/20/2021, 10/4/2020.
  The Bezout lemma is proved and used to show that the greatest common divisor (gcd) exists. Uniqueness is also shown.
  Keywords: Bezout identity, relatively prime.

• The Fundamental Theorem of Algebra and the Minimum Modulus Principle. Updated 1/28/2024, 1/15/2017, 11/8/2014, 10/11/2014, 10/2/2014.
  A direct proof of the fundamental theorem of algebra is given. In other words, we show that every polynomial of degree greater than or equal to one has at least one root in the complex plane. A slight modification of the proof yields the minimum modulus principle.

• Permutations, the Parity Theorem, and Determinants. Updated 11/3/2023, 10/25/2023, 10/15/2014, 9/14/2014, 9/12/2014, 9/11/2014, 9/2/2014.
  The Parity Theorem says that whenever an even (resp. odd) permutation is expressed as a composition of transpositions, the number of transpositions must be even (resp. odd). The purpose of this article is to give a simple definition of when a permutation is even or odd, and develop just enough background to prove the parity theorem. Several examples are included to illustrate the use of the notation and concepts as they are introduced. We then define the determinant in terms of the parity of permutations. We establish basic properties of the determinant. In particular, we show that $\det B A = \det B \, \det A$, and we show that $A$ is nonsingular if and only if $\det A \ne 0$. The characteristic polynomial is introduced and simple properties of its coefficients derived. The formula for the directional (Gateaux) derivative and total (Frechet) derivative of the determinant is also established.
  Keywords: cycle, determinant, directional derivative, even, fixed point, odd, orbit, parity, permutation, sign, signature, transposition.

• Block Matrix Formulas. Updated 7/7/2015, 7/6/2015, 7/5/2015.
  Keywords: determinant, matrix inversion formula, positive semidefinite, pseudoinverse, Sherman–Morrison–Woodbury formula, upper left block.

  Hoeffding's Inequality. Updated 11/24/2023.
  Keywords: cumulent generating function, exponentially tilted measure, Hoeffding's lemma, moment generating function, variance bound.

• Convexity Notes. Updated 3/18/2017, 10/6/2016, 8/11/2015, 8/22/2013.
  Keywords: coercive function, conjugate function, convex function, convex set, Fenchel–Legendgre transform, effective domain, epigraph, level set, lower semicontinuity, projection theorem, proper function, proximal mapping, strongly convex function, subgradient.

• Differential Equations. Updated 9/19/2014, 9/13/2014.
  Keywords: Abel/Liouville identity, adjoint equation, bilinear concomitant, companion matrix, contraction mapping theorem, existence, fixed point, fundamental matrix, Green's formula, Lagrange identity, linear differential equation, nonlinear differential equation, Sturm–Liouville, uniqueness, variation of parameters, Wronskian.

• Closable Operators. Updated 10/12/2014.
  A simple lemma is proved giving sufficient conditions for an operator to be closable.
  Keywords: closable operator, closed operator, extension.