part 1 公式與定義總整理

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1、Part1公式與定義總整理(1)Series,Integral,andTransform(非常重要)把握不同transform之間的「關聯性」，多比較彼此之間相同或相異的地方(1)LaplaceTransform(2)Fourierseries(standardform)interval:xÎ[-p,p],,,,a0,an,bn:Fouriercoefficients(2-1)Fourierseries(halfrangeextensionform)interval:xÎ[0,L]將Fourierseries的p變成L/2變成(3)Fouriercosi

2、neseries(cosineseries),適用情形：(1)interval:xÎ[-p,p],f(x)=f(-x)(2)interval:xÎ[0,p](halfrangeextension時)(4)Fouriersineseries(sineseries)適用情形：(1)interval:xÎ[-p,p],f(x)=-f(-x)(2)interval:xÎ[0,p](halfrangeextension時)(5)Fourierintegral(Sec.14-3)(6)Fouriercosineintegral(或cosineintegral),(7

3、)Fouriersineintegral(或sineintegral),(8)Fouriertransform(Sec.14-4)(8-1)inverseFouriertransform(9)Fouriercosinetransform(9-1)inverseFouriercosinetransform(10)Fouriersinetransform(10-1)inverseFouriersinetransform(2)和LaplaceTransform相關的公式(很重要)LaplacetransformDifferentiationMultiplica

4、tionbytIntegrationMultiplicationbyexpTranslation(I)Translation(II)Convolutionpropertyconvolution:PeriodicinputIff(t)=f(t+T)L{1}=1/sL{u(t)}=1/sL{tn}=L{exp(at)}=L{sin(kt)}=L{cos(kt)}=L{sinh(kt)}=L{cosh(kt)}=L{u(t−t0)}=L{d(t)}=1(3)Chapter6的相關公式與定義(i)ordinarypoint(ii)regularsingularp

5、oint(iii)irregularsingularpoint先將DE變成standardform：(i)若P0(x),P1(x),….Pn-1(x),在x=x0這一點為analytic,則x0為ordinarypoint(ii)若P0(x),P1(x),….Pn-1(x),在x=x0不為analytic,但(x-x0)nP0(x),(x-x0)n-1P1(x),......(x-x0)Pn-1(x)在x=x0為analytic,則x0為regularsingularpoint(iii)以上二條件皆不滿足，則x0為irregularsingularpoi

6、ntregularsingularpoint的情形下，r2−r1=integer時，有時(並非所有情況)要用這個式子求y2(x)Bessel’sequationofordervLegendre’sequationofordernSolutionsofBessel’sequationofordervc1Jv(x)+c2Yv(x)(4)Chapter7的相關公式與定義Stepfunctionu(t–a)=1fort>a,u(t–a)=0fort

7、ingpropertyford(t-t0)Relationbetweend(t-t0)andu(t)(5)Chapter11的相關公式與定義innerproduct*:conjugateorthogonalsquarenormnorminnerproductwithweightfunctionorthogonalwithrespecttoaweightfunctionnormalizey(x)註：orthogonalsetform¹n,noconstraintfororthonormalsetform¹n,orthogonalseriesexpansio

8、nwhereinnerproductsevenandoddIff(x)iseve