工程数学公式手册（俞超凡）.pdf

工 程 數 學 公式手冊 喻超凡博士編著 版權聲明 嚴禁以任何方式做全部或局部的改寫及轉載 ChainRule FunctionalDependence Leibnitz Gamma Beta UnitStepFunctionDeltaFunction Exact Picard ODE AODE BCD EnormalODEF GODEH IGJK LMJK I IINOPQRST UVWXJKy p LD Rx YZ GODE DODE dierence ODE ODE abcAd efg hijg jgklBjgmn TayloreBMaclaurine opqabr ODEsBts suvJODE Etwsuv xBesselyBesselzb Legendrezb SturmLiouville F SturmLiouville Fouriere Fourierab Fouriere Fourier Laplace STNOPQRIII KLaplace Duhamel LagrangePDE Z Z Z ZLaplace K HBC LaplaceJPDE FourierJPDE PDE nF VectorSets Conservativeeld B VW K Determinant J n IVNOPQRST Xn BZ RankF D LU K HGDODE HGDODE HGDODE J JordanJ CayleyHamilton nBF Quadratic HermitianK KAXBXF Cauchys Taylore Laurentse Residue cossin Fourier s sBranchpoint Laplace STNOPQRV Mapping ob NewtonRaphsonJ Lagrangian EulerCauchyODE RungeKuttaODE AdamsODE lim xa fx jxajjfxj lim xa fx MjxajjfxjM lim x fx KjxjKjfxj lim x fx MKjxjKjfxjM gxfxhxjxaj lim xa gxlim xa hxLlim xa fxL gxfxhxxk lim x gxlim x hxLlim x fxL LHospitalRule lim xa fxlim xa gxlim xa f x g x lim xa fx gx NOPQR lim xa fxlim xa gxlim xa f x g x lim xa fx gx Bolzano fxmnabfafbcabfc fxmnabfafbrfaBfbn cabfcr fxmnabvmnabfafb cabf c h fxmnabvmnabcab f c fbfa ba h fxxabcab Z b a fxdxfcba Z ah a fxdxfahh Z b a fxgxdxfc Z b a gxdxcab FundamentalTheoremoftheCalculus fab Fx Z x a ftdtxab F xfxxab Gxfx Z b a fxdxGbGaGx b a NOPQR B u a x au a u xaR e ux e ux u x a ux a ux lnau xaR lnux ux u x log a ux ux log a eu x sinux cosuxu x tanux sec uxu x secux secuxtanuxu x cosux sinuxu x cotux csc uxu x cscux cscuxcotuxu x sinhx e x e x coshx e x e x sinhux coshuxu x coshux sinhuxu x tanhux sech uxu x cothux csch uxu x sechux sechuxtanhuxu x cschux cschuxcothuxu x B sin ux a p a u u x tan ux a a a u u x NOPQR sec ux a a u p u a u x sinh ux a p a u u x cosh ux a p u a u x tanh ux a a a u u x juxj juxj ux u x ux juxj u xux Z x n dx x n n cn Z x dxlnjxjc Z e x dxe x c Z a x dx a x lna ca x log a ec Z sinxdxcosxc Z cosxdxsinxc Z sec xdxtanxcx nnZ Z csc xdxcotxcxnnZ Z secxtanxdxsecxcx nnZ Z cscxcotxdxcscxcxnnZ Z sinhxdxcoshxc NOPQR Z coshxdxsinhxc Z sech xdxtanhxc Z csch xdxcothxc Z sechxtanhxdxsechxc Z cschxcothxdxcschxc Z a x dx a tan x a c Z a x dx a tanh x a c Z p a x dxsin x a c Z x p x a dx a sec x a c Z p x a dxsinh x a c Z p x a dxcosh x a c a x a xasin dxacosd a x a cos cos a x a xatan dxasec d a x a sec sec NOPQR x a a xasec dxasectand x a a tan tan Z fsinxcosxdxf ztan x dx z dz sinx z z cosx z z uvmnI Z udvuv Z vdu NOTE Z e ax cosbxdx a b ae ax cosbxbe ax sinbxc Z e ax sinbxdx a b ae ax sinbxbe ax cosbxc Z x cosnxdxx sinnx n x cosnx n sinnx n c sinnx n H H H H H Hj cosnx n x H H H H H Hj sinnx n x H H H H H Hj cosnx Z sin n xdxB Z cos n xdx int iinsin x cosx cos x cosx NOPQR Z tan n xdx Z cot n xdxn itan xsec xcot xcsc x ii iii iva Z tanxdxlnjcosxjc b Z tan xdxtanxxc c Z cotxdxlnjsinxjc d Z cot xdxcotxxc Z tan n xdx n tan n x Z tan n xdx Z cot n xdx n cot n x Z cot n xdx Z sec n xdx Z csc n xdxn isec xtan xcsc xcot x ii iii iva Z secxdxlnjsecxtanxjc b Z sec xdxtanxc c Z cscxdxlnjcscxcotxjc d Z csc xdxcotxc Z sec n xdx n sec n xtanx n n Z sec n xdx Z csc n xdx n csc n xcotx n n Z csc n xdx Z sin m xcos n xdx imBnt iimBnh NOPQR Z tan m xsec n xdx imtn atan xsec x bdsecxsecxtanxdx iinm asec xtan x bdtanxsec xdx Z sinmxcosnxdx Z sinmxsinnxdx Z cosmxcosnxdx isinmxcosnx sinmnxsinmnx iicosmxcosnx cosmnxcosmnx iiisinmxsinnx cosmnxcosmnx Z P n x ax bxc dxP n xn b ac inP n xk xk uvJ iin AuvJ b ac inP n xk Z k ax bxc dxk Z du A u k A tan u A k iinP n xk xk Z k xk ax bxc dx k a Z axb ax bxc dxk k b a Z ax bxc dx iiin AiiiJ BCDChainRuleNOPQR Z p a x dxsin x a c Z p x a dxsinh x a clnjx p x a jc Z p x a dxcosh x a clnjx p x a jc Z P n x p ax bxc dxP n xn inP n xk Z k p ax bxc dxk Z du p A u iinP n xk xk Z k xk p ax bxc dx k a Z axb p ax bxc dx EEEEEEEEEk k b a Z p ax bxc dx iiin a Z P n x p ax bxc dxP n x p ax bxck Z dx p ax bxc EEEP n xIJn EEEkIG bFiGAHIGJP n xk Z x m abx n r dxmnr irxt mnJ ii m n abx n t p prJ iii m n rabx n t p x n prJ KLMChainRule ffxyzxxtyytzzt df dt f x dx dt f y dy dt f z dz dt ffxyzyyxzzx df dx f x f y dy dx f z dz dx ffxyzxxstyystzzst NOPQR f s f x x s f y y s f z z s f t f x x t f y y t f z z t ffxyzzzxy f x y f x f z z x f y x f y f z z y f x y yNOfx f y x xNOfy PQRSTUFunctionalDependence VWXu u hxyu u xyu u xyu Yu u Bu ZFunctionallyDependent VXnu u u n nx x x n mR u u u n J u u u n x x x n LeibnitzLM D n ffxgxg n X m C n m f nm g m Leibnitz VXfxt f x axbAtBA xB xm nab d dx Z Bx Ax fxtdt Z Bx Ax fxt x dtfxB dB dx fxA dA dx VXfxt f x axbAtA xmnab OMxtj f x jMxt Z Ax Mxtdtjg d dx Z Ax fxtdt Z Ax fxt x dtfxA dA dx GammaNOPQR GammaPQ x Z e t t x dtx x Z ln x d x Z e x d aK p xxxx xkxkxkxxxkx kk Gammax Z e t t x dtx x x xxZ p BetaPQ Bpq Z x p x q dxpq Bpqa q Z y p ya pq dy Bpq Z cos p sin q d aK BpqBqp Bpq pq pq NOPQR