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| colorinvertable='true' /> | |
| <plaintext>min{sin(x)} = -1 at x = 2 π n - π/2 for integer n</plaintext> | |
| </subpod> | |
| <subpod title=''> | |
| <img src='https://www5a.wolframalpha.com/Calculate/MSP/MSP490323h005d7ihcg97g2000029i28c1ahcd696a7?MSPStoreType=image/gif&s=24' | |
| alt='min{sin(x)} = -1 at x = 2 π n + (3 π)/2 for integer n' | |
| title='min{sin(x)} = -1 at x = 2 π n + (3 π)/2 for integer n' | |
| width='312' | |
| height='36' | |
| type='Default' | |
| themes='1,2,3,4,5,6,7,8,9,10,11,12' | |
| colorinvertable='true' /> | |
| <plaintext>min{sin(x)} = -1 at x = 2 π n + (3 π)/2 for integer n</plaintext> | |
| </subpod> | |
| <expressiontypes count='2'> | |
| <expressiontype name='Default' /> | |
| <expressiontype name='Default' /> | |
| </expressiontypes> | |
| <states count='2'> | |
| <state name='Approximate forms' | |
| input='GlobalMinimum__Approximate forms' /> | |
| <state name='Step-by-step solution' | |
| input='GlobalMinimum__Step-by-step solution' | |
| stepbystep='true' /> | |
| </states> | |
| </pod> | |
| <pod title='Alternative representations' | |
| scanner='MathematicalFunctionData' | |
| id='AlternativeRepresentations:MathematicalFunctionIdentityData' | |
| position='1200' | |
| error='false' | |
| numsubpods='3'> | |
| <subpod title=''> | |
| <img src='https://www5a.wolframalpha.com/Calculate/MSP/MSP490423h005d7ihcg97g200000h121iie41i1758g?MSPStoreType=image/gif&s=24' | |
| alt='sin(x) = 1/csc(x)' | |
| title='sin(x) = 1/csc(x)' | |
| width='94' | |
| height='46' | |
| type='Default' | |
| themes='1,2,3,4,5,6,7,8,9,10,11,12' | |
| colorinvertable='true' /> | |
| <plaintext>sin(x) = 1/csc(x)</plaintext> | |
| </subpod> | |
| <subpod title=''> | |
| <img src='https://www5a.wolframalpha.com/Calculate/MSP/MSP490523h005d7ihcg97g200005b65eife3i0fc50a?MSPStoreType=image/gif&s=24' | |
| alt='sin(x) = cos(π/2 - x)' | |
| title='sin(x) = cos(π/2 - x)' | |
| width='119' | |
| height='39' | |
| type='Default' | |
| themes='1,2,3,4,5,6,7,8,9,10,11,12' | |
| colorinvertable='true' /> | |
| <plaintext>sin(x) = cos(π/2 - x)</plaintext> | |
| </subpod> | |
| <subpod title=''> | |
| <img src='https://www5a.wolframalpha.com/Calculate/MSP/MSP490623h005d7ihcg97g200000hei26fce8855hf4?MSPStoreType=image/gif&s=24' | |
| alt='sin(x) = -cos(π/2 + x)' | |
| title='sin(x) = -cos(π/2 + x)' | |
| width='128' | |
| height='39' | |
| type='Default' | |
| themes='1,2,3,4,5,6,7,8,9,10,11,12' | |
| colorinvertable='true' /> | |
| <plaintext>sin(x) = -cos(π/2 + x)</plaintext> | |
| </subpod> | |
| <expressiontypes count='3'> | |
| <expressiontype name='Default' /> | |
| <expressiontype name='Default' /> | |
| <expressiontype name='Default' /> | |
| </expressiontypes> | |
| <states count='1'> | |
| <state name='More' | |
| input='AlternativeRepresentations:MathematicalFunctionIdentityData__More' /> | |
| </states> | |
| <infos count='2'> | |
| <info text='csc(x) is the cosecant function'> | |
| <img src='https://www5a.wolframalpha.com/Calculate/MSP/MSP490723h005d7ihcg97g200005c2ci581e0410i30?MSPStoreType=image/gif&s=24' | |
| alt='csc(x) is the cosecant function' | |
| title='csc(x) is the cosecant function' | |
| width='195' | |
| height='18' /> | |
| <link url='http://reference.wolfram.com/language/ref/Csc.html' | |
| text='Documentation' | |
| title='Mathematica' /> | |
| <link url='http://functions.wolfram.com/ElementaryFunctions/Csc' | |
| text='Properties' | |
| title='Wolfram Functions Site' /> | |
| <link url='http://mathworld.wolfram.com/Cosecant.html' | |
| text='Definition' | |
| title='MathWorld' /> | |
| </info> | |
| <info> | |
| <link url='http://functions.wolfram.com/ElementaryFunctions/Sin/27/ShowAll.html' | |
| text='More information' /> | |
| </info> | |
| </infos> | |
| </pod> | |
| <pod title='Series representations' | |
| scanner='MathematicalFunctionData' | |
| id='SeriesRepresentations:MathematicalFunctionIdentityData' | |
| position='1300' | |
| error='false' | |
| numsubpods='3'> | |
| <subpod title=''> | |
| <img src='https://www5a.wolframalpha.com/Calculate/MSP/MSP490823h005d7ihcg97g20000223d2ba6ch9cg993?MSPStoreType=image/gif&s=24' | |
| alt='sin(x) = sum_(k=0)^∞ ((-1)^k x^(1 + 2 k))/((1 + 2 k)!)' | |
| title='sin(x) = sum_(k=0)^∞ ((-1)^k x^(1 + 2 k))/((1 + 2 k)!)' | |
| width='148' | |
| height='57' | |
| type='Default' | |
| themes='1,2,3,4,5,6,7,8,9,10,11,12' | |
| colorinvertable='true' /> | |
| <plaintext>sin(x) = sum_(k=0)^∞ ((-1)^k x^(1 + 2 k))/((1 + 2 k)!)</plaintext> | |
| </subpod> | |
| <subpod title=''> | |
| <img src='https://www5a.wolframalpha.com/Calculate/MSP/MSP490923h005d7ihcg97g20000333518hfbeief76d?MSPStoreType=image/gif&s=24' | |
| alt='sin(x)∝( sum_(k=0)^∞ (-1)^k (d^(2 k) δ(x))/(dx^(2 k)))/θ(x)' | |
| title='sin(x)∝( sum_(k=0)^∞ (-1)^k (d^(2 k) δ(x))/(dx^(2 k)))/θ(x)' | |
| width='163' | |
| height='62' | |
| type='Default' | |
| themes='1,2,3,4,5,6,7,8,9,10,11,12' | |
| colorinvertable='true' /> | |
| <plaintext>sin(x)∝( sum_(k=0)^∞ (-1)^k (d^(2 k) δ(x))/(dx^(2 k)))/θ(x)</plaintext> | |
| </subpod> | |
| <subpod title=''> | |
| <img src='https://www5a.wolframalpha.com/Calculate/MSP/MSP491023h005d7ihcg97g200002fh72048bh406344?MSPStoreType=image/gif&s=24' | |
| alt='sin(x) = 2 sum_(k=0)^∞ (-1)^k J_(1 + 2 k)(x)' | |
| title='sin(x) = 2 sum_(k=0)^∞ (-1)^k J_(1 + 2 k)(x)' | |
| width='170' | |
| height='54' | |
| type='Default' | |
| themes='1,2,3,4,5,6,7,8,9,10,11,12' | |
| colorinvertable='true' /> | |
| <plaintext>sin(x) = 2 sum_(k=0)^∞ (-1)^k J_(1 + 2 k)(x)</plaintext> | |
| </subpod> | |
| <expressiontypes count='3'> | |
| <expressiontype name='Default' /> | |
| <expressiontype name='Default' /> | |
| <expressiontype name='Default' /> | |
| </expressiontypes> | |
| <states count='1'> | |
| <state name='More' | |
| input='SeriesRepresentations:MathematicalFunctionIdentityData__More' /> | |
| </states> | |
| <infos count='5'> | |
| <info text='n! is the factorial function'> | |
| <img src='https://www5a.wolframalpha.com/Calculate/MSP/MSP491123h005d7ihcg97g200004c0f017be19b7f0c?MSPStoreType=image/gif&s=24' | |
| alt='n! is the factorial function' | |
| title='n! is the factorial function' | |
| width='171' | |
| height='18' /> | |
| <link url='http://reference.wolfram.com/language/ref/Factorial.html' | |
| text='Documentation' | |
| title='Mathematica' /> | |
| <link url='http://functions.wolfram.com/GammaBetaErf/Factorial' | |
| text='Properties' | |
| title='Wolfram Functions Site' /> | |
| <link url='http://mathworld.wolfram.com/Factorial.html' | |
| text='Definition' | |
| title='MathWorld' /> | |
| </info> | |
| <info text='θ(x) is the Heaviside step function'> | |
| <img src='https://www5a.wolframalpha.com/Calculate/MSP/MSP491223h005d7ihcg97g2000010c59697b2a111h4?MSPStoreType=image/gif&s=24' | |
| alt='θ(x) is the Heaviside step function' | |
| title='θ(x) is the Heaviside step function' | |
| width='216' | |
| height='18' /> | |
| <link url='http://reference.wolfram.com/language/ref/HeavisideTheta.html' | |
| text='Documentation' | |
| title='Mathematica' /> | |
| <link url='http://functions.wolfram.com/GeneralizedFunctions/UnitStep' | |
| text='Properties' | |
| title='Wolfram Functions Site' /> | |
| <link url='http://mathworld.wolfram.com/HeavisideStepFunction.html' | |
| text='Definition' | |
| title='MathWorld' /> | |
| </info> | |
| <info text='δ(x) is the Dirac delta function'> | |
| <img src='https://www5a.wolframalpha.com/Calculate/MSP/MSP491323h005d7ihcg97g200002e85433e829ie851?MSPStoreType=image/gif&s=24' | |
| alt='δ(x) is the Dirac delta function' | |
| title='δ(x) is the Dirac delta function' | |
| width='195' | |
| height='18' /> | |
| <link url='http://reference.wolfram.com/language/ref/DiracDelta.html' | |
| text='Documentation' | |
| title='Mathematica' /> | |
| <link url='http://functions.wolfram.com/GeneralizedFunctions/DiracDelta' | |
| text='Properties' | |
| title='Wolfram Functions Site' /> | |
| <link url='http://mathworld.wolfram.com/DeltaFunction.html' | |
| text='Definition' | |
| title='MathWorld' /> | |
| </info> | |
| <info text='J_n(z) is the Bessel function of the first kind'> | |
| <img src='https://www5a.wolframalpha.com/Calculate/MSP/MSP491423h005d7ihcg97g200002e2e43553cff291h?MSPStoreType=image/gif&s=24' | |
| alt='J_n(z) is the Bessel function of the first kind' | |
| title='J_n(z) is the Bessel function of the first kind' | |
| width='278' | |
| height='18' /> | |
| <link url='http://reference.wolfram.com/language/ref/BesselJ.html' | |
| text='Documentation' | |
| title='Mathematica' /> | |
| <link url='http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ' | |
| text='Properties' | |
| title='Wolfram Functions Site' /> | |
| <link url='http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html' | |
| text='Definition' | |
| title='MathWorld' /> | |
| </info> | |
| <info> | |
| <link url='http://functions.wolfram.com/ElementaryFunctions/Sin/06/ShowAll.html' | |
| text='More information' /> | |
| </info> | |
| </infos> | |
| </pod> | |
| <pod title='Integral representations' | |
| scanner='MathematicalFunctionData' | |
| id='IntegralRepresentations:MathematicalFunctionIdentityData' | |
| position='1400' | |
| error='false' | |
| numsubpods='3'> | |
| <subpod title=''> | |
| <img src='https://www5a.wolframalpha.com/Calculate/MSP/MSP491523h005d7ihcg97g2000037i4feh0a12ai85g?MSPStoreType=image/gif&s=24' | |
| alt='sin(x) = x integral_0^1 cos(t x) dt' | |
| title='sin(x) = x integral_0^1 cos(t x) dt' | |
| width='150' | |
| height='41' | |
| type='Default' | |
| themes='1,2,3,4,5,6,7,8,9,10,11,12' | |
| colorinvertable='true' /> | |
| <plaintext>sin(x) = x integral_0^1 cos(t x) dt</plaintext> | |
| </subpod> | |
| <subpod title=''> | |
| <img src='https://www5a.wolframalpha.com/Calculate/MSP/MSP491623h005d7ihcg97g200003bh805d217ca2213?MSPStoreType=image/gif&s=24' | |
| alt='sin(x) = -(i x)/(4 sqrt(π)) integral_(-i ∞ + γ)^(i ∞ + γ) e^(s - x^2/(4 s))/s^(3/2) ds for γ>0' | |
| title='sin(x) = -(i x)/(4 sqrt(π)) integral_(-i ∞ + γ)^(i ∞ + γ) e^(s - x^2/(4 s))/s^(3/2) ds for γ>0' | |
| width='289' | |
| height='55' | |
| type='Default' | |
| themes='1,2,3,4,5,6,7,8,9,10,11,12' | |
| colorinvertable='true' /> | |
| <plaintext>sin(x) = -(i x)/(4 sqrt(π)) integral_(-i ∞ + γ)^(i ∞ + γ) e^(s - x^2/(4 s))/s^(3/2) ds for γ>0</plaintext> | |
| </subpod> | |
| <subpod title=''> | |
| <img src='https://www5a.wolframalpha.com/Calculate/MSP/MSP491723h005d7ihcg97g200000d65b6198f68gfa2?MSPStoreType=image/gif&s=24' | |
| alt='sin(x) = -i/(2 sqrt(π)) integral_(-i ∞ + γ)^(i ∞ + γ) (2^(-1 + 2 s) x^(1 - 2 s) Γ(s))/Γ(3/2 - s) ds for (0<γ<1 and x>0)' | |
| title='sin(x) = -i/(2 sqrt(π)) integral_(-i ∞ + γ)^(i ∞ + γ) (2^(-1 + 2 s) x^(1 - 2 s) Γ(s))/Γ(3/2 - s) ds for (0<γ<1 and x>0)' | |
| width='431' | |
| height='58' | |
| type='Default' | |
| themes='1,2,3,4,5,6,7,8,9,10,11,12' | |
| colorinvertable='true' /> | |
| <plaintext>sin(x) = -i/(2 sqrt(π)) integral_(-i ∞ + γ)^(i ∞ + γ) (2^(-1 + 2 s) x^(1 - 2 s) Γ(s))/Γ(3/2 - s) ds for (0<γ<1 and x>0)</plaintext> | |
| </subpod> | |
| <expressiontypes count='3'> | |
| <expressiontype name='Default' /> | |
| <expressiontype name='Default' /> | |
| <expressiontype name='Default' /> | |
| </expressiontypes> | |
| <infos count='2'> | |
| <info text='Γ(x) is the gamma function'> | |
| <img src='https://www5a.wolframalpha.com/Calculate/MSP/MSP491823h005d7ihcg97g200004dh85520h4d848d8?MSPStoreType=image/gif&s=24' | |
| alt='Γ(x) is the gamma function' | |
| title='Γ(x) is the gamma function' | |
| width='176' | |
| height='18' /> | |
| <link url='http://reference.wolfram.com/language/ref/Gamma.html' | |
| text='Documentation' | |
| title='Mathematica' /> | |
| <link url='http://functions.wolfram.com/GammaBetaErf/Gamma' | |
| text='Properties' | |
| title='Wolfram Functions Site' /> | |
| <link url='http://mathworld.wolfram.com/GammaFunction.html' | |
| text='Definition' | |
| title='MathWorld' /> | |
| </info> | |
| <info> | |
| <link url='http://functions.wolfram.com/ElementaryFunctions/Sin/07/ShowAll.html' | |
| text='More information' /> | |
| </info> | |
| </infos> | |
| </pod> | |
| <pod title='Definite integral over a half-period' | |
| scanner='InterestingDefiniteIntegrals' | |
| id='DefiniteIntegralOverAHalfPeriod' | |
| position='1500' | |
| error='false' | |
| numsubpods='1'> | |
| <subpod title=''> | |
| <img src='https://www5a.wolframalpha.com/Calculate/MSP/MSP491923h005d7ihcg97g20000214fi1a3fic6gg59?MSPStoreType=image/gif&s=24' | |
| alt='integral_0^π sin(x) dx = 2' | |
| title='integral_0^π sin(x) dx = 2' | |
| width='103' | |
| height='33' | |
| type='Default' | |
| themes='1,2,3,4,5,6,7,8,9,10,11,12' | |
| colorinvertable='true' /> | |
| <plaintext>integral_0^π sin(x) dx = 2</plaintext> | |
| </subpod> | |
| <expressiontypes count='1'> | |
| <expressiontype name='Default' /> | |
| </expressiontypes> | |
| </pod> | |
| <pod title='Definite integral mean square' | |
| scanner='InterestingDefiniteIntegrals' | |
| id='DefiniteIntegralMeanSquare' | |
| position='1600' | |
| error='false' | |
| numsubpods='1'> | |
| <subpod title=''> | |
| <img src='https://www5a.wolframalpha.com/Calculate/MSP/MSP492023h005d7ihcg97g200002bb00a8600a3i444?MSPStoreType=image/gif&s=24' | |
| alt='integral_0^(2 π) (sin^2(x))/(2 π) dx = 1/2 = 0.5' | |
| title='integral_0^(2 π) (sin^2(x))/(2 π) dx = 1/2 = 0.5' | |
| width='165' | |
| height='39' | |
| type='Default' | |
| themes='1,2,3,4,5,6,7,8,9,10,11,12' | |
| colorinvertable='true' /> | |
| <plaintext>integral_0^(2 π) (sin^2(x))/(2 π) dx = 1/2 = 0.5</plaintext> | |
| </subpod> | |
| <expressiontypes count='1'> | |
| <expressiontype name='Default' /> | |
| </expressiontypes> | |
| </pod> | |
| </queryresult> |
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