delta function mathematics delta function function function distribution real number line zero zero integral entire real line delta function thought hypothetical function infinitely high infinitely thin spike origin total area spike physically represents density point mass point charge introduced theoretical physicist purely mathematical viewpoint delta strictly function extended real function equal zero single point total integral zero delta function makes sense mathematical object appears inside integral perspective delta usually manipulated function formal rules function operational standard tool kit physics engineering operational particular delta function viewed suspicion early 20th century satisfactory rigorous theory introduced 1950s formally delta function defined distribution corresponds probability measure supported origin applications delta regarded kind limit weak limit sequence functions having tall spike origin functions sequence approximate nascent delta functions context signal processing delta function referred unit impulse symbol function analog delta function usually defined domain takes values overview delta function usually thought following axis positive axis despite delta function truly function usual range real numbers example objects equal different according integration theory functions identical rigorous treatment delta requires measure theory theory delta model tall narrow spike function impulse similar point charge point mass electron point example dynamics baseball hit bat approximate force bat hitting baseball delta function able motion baseball considering total impulse bat ball requiring knowledge details bat transferred energy ball applied mathematics delta function manipulated kind limit weak limit sequence functions member tall spike origin example sequence centered origin tending zero history presented called integral treatise form formula introduction function form formula later expressed formula pointed circumstances order integration result significant justified theory equation rearranged resemble original expose function formula function expressed formula rigorous interpretation form limitations function necessary application extended centuries problems classical interpretation explained follows greatest classical transformation narrow class functions originals effectively necessary functions decrease sufficiently rapidly zero neighborhood infinity order ensure existence integral example transform simple functions exist classical sense extension classical transformation considerably enlarged class functions transformed removed obstacles developments included integral beginning theory continuing works culminating amalgamation theory leading formal development delta function formula infinitely tall unit impulse delta function version distribution explicitly appears text considered issue connection study wave propagation somewhat later introduced unit impulse limit corresponded lord notion point heat source end 19th century formal series manipulate unit impulse delta function introduced convenient influential book principles called delta function continuous delta definitions delta loosely thought function real line zero origin infinite formula constrained satisfy identity formula merely characterization delta function traditional sense function defined real numbers properties delta function defined distribution measure measure way capture notion delta function define measure accepts argument real line returns delta function modeling point mass represents mass contained set define integral integral function mass distribution formally integral provides necessary device integral respect measure formula continuous supported functions measure absolutely continuous respect measure fact singular measure consequently delta measure derivative true function property formula holds result convenient abuse standard integral probability measure delta measure characterized cumulative distribution function unit step function formula means integral cumulative indicator function respect measure wit formula particular integral delta function continuous function properly understood integral formula higher moments zero particular characteristic function moment generating function equal distribution theory function thought function relation affects functions integrated keeping philosophy define delta function properly integral delta function sufficiently good test function delta function understood measure integral test function measure supplies necessary integral typical space test functions consists smooth functions compact support distribution delta linear functional space test functions defined test function properly distribution continuous suitable space test functions general linear functional space test functions define distribution necessary sufficient positive constant test function inequality formula distribution inequality distribution order zero furthermore distribution compact support support delta distribution defined number equivalent ways instance derivative step function means test function formula integration parts permitted integral formula form integration parts permitted integral case formula context measure theory measure gives rise distribution integration conversely equation defines integral space supported continuous functions representation represented integral respect measure generally term delta function sense measures measure terms corresponding notion measure theory sources use term delta distribution delta function defined dimensional space measure formula supported continuous function measure dimensional delta function product measure dimensional delta functions variable separately formally delta function defined sense exactly dimensional case despite widespread use engineering contexts manipulated care product defined narrow circumstances notion measure makes sense set set marked point measure defined sets formula delta measure unit mass concentrated common delta function properties distribution exploited structure delta function centered point defined following distribution supported smooth real valued functions common special case construction open set space locally compact space delta measure concentrated point measure associated integral supported continuous functions level longer possible variety techniques abstract analysis available instance mapping formula continuous space finite measures equipped vague convex hull image dense space probability measures properties scaling symmetry delta function following scaling property non zero formula particular delta function distribution sense formula degree properties product equal zero formula formula constant translation integral time delayed delta given formula referred property sampling property delta function said value follows effect function time delayed delta time delay holds precise condition tempered distribution discussion transform special case instance identity understood distribution sense formula composition function generally delta distribution composed smooth function way familiar change formula holds formula provided continuously function zero unique way assign meaning distribution formula identity holds supported test functions domain broken exclude point distribution zero real root formula natural define composition continuously functions formula sum extends roots assumed simple example formula integral form scaling property written formula properties dimensions delta distribution dimensional space following scaling property instead formula distribution degree reflection rotation delta function formula variable case possible define composition function uniquely identity formula supported functions formula geometric measure theory define composition delta function space different dimension result type current special case continuously function gradient zero following identity holds formula integral right surface defined respect content measure known simple layer integral generally smooth associate distribution supported smooth function formula measure associated associated potential theory simple layer domain smooth boundary equal normal derivative indicator function distribution sense formula outward normal proof article surface delta function transform delta function tempered distribution defined transform formally finds formula properly speaking transform distribution defined imposing self transform pairing formula tempered functions formula defined unique tempered distribution satisfying formula functions follows formula result identity delta function tempered distribution simply formula identity element tempered fact space supported identity delta function property fundamental signal processing tempered distribution linear time system applying linear time system measures impulse response impulse response desired degree accuracy choosing suitable known characterizes system completely system theory impulse response transform tempered distribution delta function formally expressed formula follows formula functions terms delta function provides suggestive statement property formula course assertion transform tempered distribution formula formula follows imposing self transform continuation transform transform delta function found formula derivatives derivative delta distribution distribution defined supported smooth test functions formula equality kind integration parts true function formula derivative defined similarly distribution given test functions formula particular infinitely distribution derivative delta function limit difference formula properly formula translation operator defined functions distribution formula theory derivative delta function represents point magnetic situated origin accordingly referred function derivative delta function number basic properties including formula furthermore supported smooth function formula follows properties derivative higher dimensions generally open set dimensional space delta distribution centered point defined formula space smooth supported functions multi index denotes associated mixed partial derivative operator derivative given formula derivative distribution value test function derivative appropriate positive negative sign partial derivatives delta function thought double layers coordinate planes generally normal derivative simple layer supported surface double layer supported surface represents magnetic higher derivatives delta function known physics higher derivatives enter mathematics naturally building blocks complete structure point support distribution supported set consisting single point formula representations delta function delta function viewed limit sequence functions formula called nascent delta function limit meant weak sense continuous functions having compact support limit holds smooth functions compact support difference slightly different modes weak convergence subtle convergence vague measures convergence sense identity typically nascent delta function constructed following manner let absolutely function total integral define formula dimensions uses instead scaling formula simple change shows integral shows easily holds continuous supported functions weakly sense measures constructed way known identity terminology space absolutely functions closed operation functions identity product element sequence approximate identity sense formula limit holds sense mean convergence convergence conditions instance associated supported function needed ensure convergence initial smooth supported sequence called standard obtained choosing suitably normalized bump function instance formula situations numerical analysis linear identity desirable obtained taking hat function choice formula continuous supported smooth considerations context probability theory natural impose additional condition initial identity positive function represents probability distribution probability distribution favorable result output convex combination input values falls maximum minimum input function taking probability distribution letting rise identity general rapidly delta function addition mean small higher moments instance uniform distribution known rectangular function formula example distribution formula continuous supported smooth nascent delta functions arise amounts satisfy formula form nascent delta function identity sense condition strong restriction practice delta function arise fundamental solutions green functions physically motivated partial differential context applied mathematics arise output linear time system linear operator acting functions arises solving initial value problem formula limit usual understood weak sense setting gives associated nascent delta function examples physically important arising fundamental solution include following heat heat defined formula represents temperature infinite wire time unit heat energy stored origin wire time according dimensional heat equation formula probability theory normal distribution mean represents probability density time position particle starting origin following standard motion context condition expression property motion higher dimensional space heat formula physical interpretation represents nascent delta function sense distribution sense formula fundamental solution equation upper half plane represents potential semi infinite plate potential edge held fixed delta function closely related distribution according equation formula operator defined formula areas physics wave propagation wave mechanics involved singular solutions result nascent delta functions arise fundamental solutions associated problems generally example comes solution equation gas dynamics function formula transform easy generates sense absolutely define strong sense nascent delta functions constructed converge sense example sense measures example problem wave equation formula solution represents displacement infinite elastic string initial disturbance origin identity kind include function widely electronics telecommunications formula function formula plane wave approach study linear partial differential equation formula differential operator seek fundamental solution solution equation formula particularly simple problem resolved transform directly case heat mentioned complicated operators easier consider equation form formula plane wave function meaning form formula equation resolved functions constant delta function plane waves principle solve linear partial differential delta function plane waves general technique introduced essentially developed form choose real number formula obtained applying power integral respect unit sphere measure unit sphere formula interpreted weak derivative equation taken mean test function formula result follows formula potential fundamental solution equation essentially form formula transform recovers value instance odd integral right hand formula transform formula alternative equivalent expression plane wave formula formula odd study series major question consists determining sense series associated periodic function function partial sum series function period defined interval formula formula formula fundamental result elementary series states tends multiple delta function interpreted distribution sense formula supported smooth function formally formula interval spite result hold supported continuous functions converge weakly sense measures lack convergence series led introduction variety methods order produce convergence method leads formula tend delta function stronger sense formula supported continuous function series continuous function value function point space theory delta distribution densely defined linear functional space square functions smooth support functions dense action delta distribution functions defined applications possible identify stronger delta function defines bounded linear functional spaces spaces real line implies square function formula automatically continuous particular formula bounded linear functional space element continuous dual space generally dimensions provided spaces functions complex analysis delta function enters integral formula asserts domain complex plane smooth boundary formula functions continuous closure result delta function represented class functions integral formula generally let hardy space consisting closure functions continuous boundary functions uniquely extend functions integral formula continues hold particular delta function continuous linear functional special case situation complex smooth domains plays role integral resolutions identity given complete basis set functions space example normalized compact self operator expressed formula found formula represented formula form bra adopting expansion takes form formula letting denote identity operator space expression formula called resolution identity space space square functions domain quantity formula integral operator expression rewritten formula right hand sense need hold sense continuous function common abuse write formula resulting representation delta function formula suitable rigged space contains supported smooth functions converge depending properties basis cases practical interest basis comes integral differential operator case series distribution sense delta functions write unit impulse infinitely tall narrow type delta function satisfying formula number articles defined terms sequence tending zero sequence terminology non standard analysis allows treat article contains modern delta functions context enriched provided delta given actual function having property real function formula anticipated called uniform pulse train delta measures known distribution creates sampling function digital signal processing time signal analysis given infinite sum limit understood distribution sense formula sequence point masses overall constant equal transform significant function given formula particular formula precisely formula important quantum mechanics relates delta function distribution principal value function defined formula formula states formula limit understood distribution sense supported smooth functions formula relationship delta delta quantity defined formula function following analog property formula infinite sequence formula similarly real complex valued continuous function delta property formula exhibits delta function analog delta function applications probability theory probability theory statistics delta function represent distribution partially partially continuous distribution probability density function normally represent fully continuous example probability density function distribution consisting points corresponding written formula example consider distribution time returns standard normal distribution time returns exactly value partly continuous partly mixture distribution density function distribution written formula delta function completely different way represent local time process like motion local time process given formula represents time process spends point range process precisely dimension integral written formula indicator function interval quantum mechanics example delta function quantum mechanics wave function particle gives probability finding particle given region space wave functions assumed elements space square functions total probability finding particle given interval integral magnitude wave function squared interval set wave functions normalized formula refers delta set wave functions complete space square functions wave function expressed combination formula formula complete systems wave functions appear naturally bound system quantum mechanics measures energy levels called set case known spectrum bra equality implies resolution identity formula assumed set continuous example position spectrum position dimension entire real line called continuous spectrum unlike position operator lacks proper conventional way overcome widen class available functions allowing replace space quantum mechanics appropriate rigged space context position operator complete set labeled points real line given formula position denoted formula known position similar considerations apply momentum operator self operator space provided spectrum continuous case set real numbers spectrum collection elements formula normalized formula distribution sense test function formula formula case resolution identity formula operator valued integral understood weak sense spectrum continuous parts resolution identity involves spectrum integral continuous spectrum delta function specialized applications quantum mechanics delta potential models single double potential structural mechanics delta function structural mechanics describe loads point loads acting structures governing equation simple mass spring system excited sudden force impulse time written formula mass spring constant example equation governing static slender beam according theory formula bending beam spatial coordinate load distribution beam loaded point force load distribution written formula integration delta function results step function follows static slender beam subject multiple point loads described set point moment acting beam described delta functions consider opposing point forces distance apart produce moment acting beam let distance approach limit zero kept constant load distribution assuming clockwise moment acting written formula point moments represented derivative delta function integration beam equation results
Dirac delta function