# Index [[2-norm]] [[Almost everywhere convergence]] [[analytic function]] [[Boxcar function]] [[bump function]] [Cauchy principal value](Cauchy%20principal%20value.md) [Complete sequence](Complete%20sequence.md) [[Continuous function]] [[Convergence theorems]] [[Convolution]] [Differentiability of a functional](Differentiability%20of%20a%20functional.md) [Differential operator](Differential%20operator.md) [[Dirac delta function]] [[Dirichlet integral]] [[Dirichlet's principle]] [[Discontinuities]] [[Distribution]] [[Elliptical integral]] [[Exponential integrals]] [Fresnel integral](Fresnel%20integral.md) [[Gamma function]] [[Gaussian function]] [[Gaussian integral]] [[Gaussian representation of 𝛿(y-x)]] [Gauss's theorem](Gauss's%20theorem.md) [[Generalized functions]] [[Heaviside function]] [[Hermite functions]] [[Hermite polynomial]] [[Hölder's inequality]] [[Hyperbolic representation of 𝛿(y-x)]] [[Improper integrals]] [Infinitessimal number](Infinitessimal%20number.md) [[Infinum of a function]] [[Integral transform]] [[Laplace operator]] [[Lebesgue integral]] [[Lebesgue's dominated convergence theorem]] [[Legendre-Fenchel transform]] [[Legendre Polynomials]] [[Legendre transformation]] [Length of a curve](Length%20of%20a%20curve.md) [Linear differential operators](Linear%20differential%20operators.md) [[Lorentzian function]] [[Lorentzian representation of 𝛿(y-x)]] [[Monotone convergence theorem]] [[Multi-dimensional Dirac delta function]] [[periodic functions]] [[Rectangle function representation of 𝛿(y-x)]] [[Representation of 𝛿(y-x)]] [[Riemann integral]] [[scalar field]] [Sequences](Sequences.md) [[Sequences of functions]] [[single valued function]] [[Singularity]] [[Smooth function]] [[Support]] [[Supremum of a function]] [total differential of a function](total%20differential%20of%20a%20function) ## Sub-indices [[Complex analysis (index)]] [[Differential equations (index)]] [[Fourier analysis (Index)]] [[Functional analysis (index)]] [Probability distributions (index)](Probability%20distributions%20(index).md) --- # Proofs and examples [[Proof of property 8 of the Dirac delta function]] [[Proof that the Dirac delta is differentiable with a test function]] [Proof that the Dirac delta function may be represented with a Gaussian function](Proof%20that%20the%20Dirac%20delta%20function%20may%20be%20represented%20with%20a%20Gaussian%20function.md) --- # Forward _Calculus_ is the collection of algorithms and theorems that describe the use of [limits,](Analysis%20(index).md#Limits) [derivatives,](Analysis%20(index).md#Derivatives) [elementary differential equations,](Analysis%20(index).md#Differential%20equations) [sums,](Analysis%20(index).md#Sums) and [integrals](Analysis%20(index).md#Integrals) in order to manipulate [functions.](Analysis%20(index).md#Functions) More elementary concepts and methods for manipulating functions are considered part of _pre-calculus, trigonometry,_ or simply _algebra._ However, a complete and precise understanding of calculus and its preliminaries requires _analysis,_ which provides us with all the theorems that prove that calculus works and expand on why and how calculus works. ## Structure of analysis In the top level [index](Analysis%20(index).md#Index) we choose to list everything that may be considered _real Analysis_ as well as some aspects _[functional analysis](Functional%20analysis%20(index).md)_ with real numbers while _[complex analysis,](Complex%20analysis%20(index).md)_ deals with functions containing _[complex numbers](Complex%20analysis%20(index).md#Numbers%20on%20a%20complex%20plane)_, and _[Fourier analysis](Fourier%20analysis%20(Index).md)_ deals with _[trigonometric functions](Analysis%20(index).md#Trigonometric%20Functions)_ and more broadly _[periodic functions.](periodic%20functions)_ An important intersection between [linear algebra](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md) and analysis is _[vector analysis.](Vector%20analysis%20(index).md)_ In addition [functional analysis](Functional%20analysis%20(index).md) may also be viewed as an extension of linear algebra, since sets of functions form [[Infinite dimensional vector spaces]] called _[function spaces.](Function%20spaces.md)_ One may also say that functional analysis is an application of concepts from linear algebra towards understanding functions. Finally some applications of different areas of analysis are in solving _[differential Equations](Differential%20equations%20(index).md)_ as well as in understanding _[probability distributions.](Probability%20distributions%20(index).md)_ ### Harmonic analysis _Harmonic Analysis_ is a generalization of [Fourier Analysis.](Fourier%20analysis%20(Index).md) For now topics that may be labeled as part of harmonic analysis will find their way to the index for Fourier Analysis as well as this top level index on [real and functional analysis.](Analysis%20(index).md) ### Measure theory --- # Basic Concepts Below I attempt to present concepts and formulas in a format that may be thought of as a crash course or cheat sheet that covers what may be considered high school level calculus and precalculus or part of what may be covered also in a first year calculus course at a university. The topics in those courses are usually what students are exposed to before they studying analysis. ## Sums ### Power series #### Geometric Series A _Geometric Series_ converges as follows $\sum_n^{\infty}x^n=\frac{1}{1-x}$ ## Functions [[Maps]], [Continuous function](Continuous%20function.md) ### Bounded and unbounded ### Symmetric and anti-symmetric ### Piecewise functions ### Common functions #### Trigonometric functions $\sin{x}=\cos{(x-\frac{\pi}{2})}$ $\cos{x} = \sin({x+\frac{\pi}{2}})$ and $\sin{x}=-\cos{(x+\frac{\pi}{2})}$ $\cos{x} = -\sin({x-\frac{\pi}{2}}).$ ^f35292 ##### Trigonometric identities #### Jordan's inequality $\frac{2}{\pi}x\leq\sin(x)\leq x\;\;\mbox{for} \;\; x \in \bigg[0,\frac{\pi}{2}\bigg]$ where we can see this inequality graphically:  #### Exponential functions ##### As a power series The [exponential function](Analysis%20(index).md#Exponential%20functions) may be written as a [Power series.](Analysis%20(index).md#Power%20series) $e^x=\sum_{n=0}^{\infty} \frac{x^n}{n!}$ ##### As a limit of a sequence $e^x = \lim_{n\rightarrow\infty}\bigg(1+\frac{x}{n}\bigg)^n$ #### Natural log This is the inverse of the [exponential function.](Analysis%20(index).md#Exponential%20functions) ##### As an infinite series The [natural log](Analysis%20(index).md#Natural%20log) may be expressed as the following [series expansion](Analysis%20(index).md#Sums) $\ln(x)=-\sum_{n=1}^{\infty}\frac{(-1)^n(x-1)^n}{n}$ where $|x-1|<2.$ #### Hyperbolic functions $\coth(x)=\frac{\cosh{x}}{\sinh{x}}=\frac{e^{-x}+e^x}{e^x-e^{-x}}$ ##### Hyperbolic trigonometric identities $\cosh^2{x}-\sinh^2{x}=1$ ## Limits ## Derivatives ### Derivative rules #### Chain rule ### Extrema ### Taylor Series ### Differential equations ## Integrals and anti-derivatives The definition of the integral is linked to the derivative via the _fundamental theorem of calculus_, which is given as $\int_{a}^{b}dx f(x) = F(b)-F(a)$ where $F(x)$ is the _antiderivative_ of $f(x)$. The same theorem and subsequent elementary rules listed below also apply under the definition of the [Riemann integral.](Riemann%20integral.md) The limits $a$ and $b$ are any pair of real numbers. We may then also define [improper integrals.](Improper%20integrals.md) ### Elementary rules of integration ### Substitution rules ##### Trigonometric substitution For many integrals containing trigonometric functions it's convenient to express them in terms of another variable set equal to the [trigonometric function.](Analysis%20(index).md#Trigonometric%20Functions) ##### Integration by parts ### Averages of functions Integrals over intervals let us find the _average_ or _mean_ of a function. For a, function $f(x),$ the average on an interval $\Delta x = b-a$ is defined as $\bar{f}(x) = \frac{\int_a^b dx\,f(x)}{b-a}.$ %%This is from pages 347-348 of Boas and is treated as a prelude to the discussion on Fourier series.%% ### Integral equations ## Elementary algebraic manipulation For these notes we will consider Algebraic manipulation as it pertains to variables in equations to be assumed knowledge. However, there are time-saving "tricks" that are good to be reminded of that are listed below. Generally these tricks are meant to save time during lengthy derivations. ### Partial fraction decomposition ### Completing the square --- # Recommended reading For a general introduction to calculus that covers the prerequisite material needed for [analysis](Analysis%20(index).md) see: * [Strang, G., _Calculus_, Wellesley-Cambridge Press (1991).](Gilbert%20Strang%20Calculus.pdf) This text covers the full range of topics generally included in _Calculus 1_, _Calculus 2_ and parts of _Calculus 3_, courses at American universities as well as American high school equivalents to Calc 1 and Calc 2, _AP Calculus AB_ and _AP Calculus BC._ It is publicly available [here.](https://ocw.mit.edu/ans7870/resources/Strang/Edited/Calculus/Calculus.pdf) Additional resources provided by the author are given [here.](https://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/index.htm) Many of the same calculus topics needed for Analysis are included here: * [Boas M., _Mathematical Methods in the Physical Sciences_. John Wiley and Sons, 3rd edition, 2006.](Boas%20M.,%20Mathematical%20Methods%20in%20the%20Physical%20Sciences.%20John%20Wiley%20and%20Sons,%203rd%20edition,%202006..md) Chapter 1 includes a more elementary discussion of sums and infinite series that would be considered part of _Calculus 2._ The remaining text introduces a wide range of topics in analysis, complex analysis, topics in _Calculus 3_, as well as Linear Algebra that's primarily aimed at 1st and 2nd year undergraduates studying physics. For an introduction that motivates the study of [analysis](Analysis%20(index).md) as well as a basic overview of its structure see: * [Tao, T. _Analysis I_, Hindustan Book Agency & Springer (3rd edition, 2016)](TerenceTao_Analysis.I.Third.Edition.pdf) pgs. 1-12. Here there are several motivating examples from elementary algebra and calculus. Moreover the structure and scope of this text along with structure and scope of [_Analysis II_,]([Texts%20and%20Readings%20in%20Mathematics]%20Terence%20Tao%20-%20Analysis%20II%20(2016,%20Springer)%20-%20libgen.lc.pdf) which is meant as a 2nd volume to this text. These texts are aimed at advanced undergraduates in mathematics. --- # Bibliography [Altland, A. von Delft, J. _Mathematics for Physicists_, Cambridge University Press, 2019](Altland,%20A.%20von%20Delft,%20J.%20Mathematics%20for%20Physicists,%20Cambridge%20University%20Press,%202019.md) [Boas M., _Mathematical Methods in the Physical Sciences_. John Wiley and Sons, 3rd edition, 2006.](Boas%20M.,%20Mathematical%20Methods%20in%20the%20Physical%20Sciences.%20John%20Wiley%20and%20Sons,%203rd%20edition,%202006..md) [Brigham E. O., _The Fast Fourier Transform and Its Applications_, Prentice Hall, 1988.](Brigham%20E.%20O.,%20The%20Fast%20Fourier%20Transform%20and%20Its%20Applications,%20Prentice%20Hall,%201988..md) [Brown, J. W., Churchill R. V., _Complex Variables and Applications_, McGraw Hill, 8th edition, 2009.](Brown,%20J.%20W.,%20Churchill%20R.%20V.,%20Complex%20Variables%20and%20Applications,%20McGraw%20Hill,%208th%20edition,%202009..md) [Griffiths, D. J., _Introduction to Electrodynamics_, Pearson Prentice Hall, 4th edition, 2017.](Griffiths,%20D.%20J.,%20Introduction%20to%20Electrodynamics,%20Pearson%20Prentice%20Hall,%204th%20edition,%202017..md) [Hall, B., _Lie Groups Lie Algebras and Representations_, Springer, 2nd edition, 2015.](Hall,%20B.,%20Lie%20Groups%20Lie%20Algebras%20and%20Representations,%20Springer,%202nd%20edition,%202015..md) [Schollwöck, U. Homework 1, Quantum Mechanics 1 (German) (2019-2020)](Schollwöck,%20U.%20Homework%201,%20Quantum%20Mechanics%201%20(German)%20(2019-2020).md) Murayama H., MH2801: Complex Methods for the Sciences  Gradshteyn I. S., Ryzhik I. M. _Table of Integrals, Series, and Products_ .pdf) Bartle R., Sherbert D., _Introduction to Real Analysis_ .pdf) Brychkov, Y. A., Prudnikov A. P. _Integral Transforms of Generalized Functions_ .pdf) Engel, E., Dreizler, R. M., _Density Functional Theory - An Advanced Course_ %20-%20Density%20Functional%20Theory_%20An%20Advanced%20Course%20(2011,%20Springer-Verlag%20Berlin%20Heidelberg)%20-%20libgen.lc.pdf) Arken G. B., _Mathematical methods for Physicists_ %20-%20libgen.lc.pdf) Strang, G., _Calculus_  Schollwöck U, _Advanced Statistical Physics_, Lecture Notes, Advanced Statistical Physics, Chapter 9, (2021) (preprint)  Tao, T. _Analysis I_, Hindustan Book Agency & Springer (3rd edition, 2016)  Tao, T. _Analysis II_, Hindustan Book Agency & Springer (3rd edition, 2016) %20-%20libgen.lc.pdf) Courant R., Hilbert D., _Methods of Mathematical Physics_ Vol. I, %20-%20libgen.lc.pdf) [von Neumann J., _Mathematical Foundations of Quantum Mechanics_. Translated by Robert T. Beyer. Princeton University Press, 2018.](von%20Neumann%20J.,%20Mathematical%20Foundations%20of%20Quantum%20Mechanics.%20Translated%20by%20Robert%20T.%20Beyer.%20Princeton%20University%20Press,%202018..md) [Weisstein, Eric W. "Boxcar Function." From MathWorld--A Wolfram Web Resource](Weisstein,%20Eric%20W.%20Boxcar%20Function.%20From%20MathWorld--A%20Wolfram%20Web%20Resource.md) [Weisstein, Eric W. "Heaviside Step Function." From MathWorld--A Wolfram Web Resource.](Weisstein,%20Eric%20W.%20Heaviside%20Step%20Function.%20From%20MathWorld--A%20Wolfram%20Web%20Resource..md) #MathematicalFoundations/Analysis #MathematicalFoundations/Analysis/Functions #MathematicalFoundations/Analysis/Integrals #MathematicalFoundations/Analysis/Functionals/Integrals/IntegralTransforms #MathematicalFoundations/Analysis/GeneralizedFunctions #MathematicalFoundations/Analysis/Functions/Functionals #MathematicalFoundations/Analysis/Derivatives #MathematicalFoundations/Analysis/ComplexAnalysis #MathematicalFoundations/Analysis/ComplexAnalysis/Integrals #MathematicalFoundations/Analysis/ComplexAnalysis/Functions #MathematicalFoundations/Analysis/DifferentialEquations/OrdinaryDifferentialEquations #MathematicalFoundations/Analysis/DifferentialEquations/PartialDifferentialEquations #MathematicalFoundations/Analysis/FourierAnalysis #MathematicalFoundations/Analysis/FourierAnalysis/Integrals #MathematicalFoundations/Analysis/VectorAnalysis #MathematicalFoundations/Analysis/VectorAnalysis/Derivatives #MathematicalFoundations/Analysis/VectorAnalysis/Integrals #MathematicalFoundations/Analysis/FunctionalAnalysis #Bibliography #index