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Gauss hypergeometric equation x(1-x)y"+[c-(a+b+1)x]y'-aby=0
Gauss hypergeometric equation solution 1+abx/1!c+a(a+1)b(b+1)x^2/2!c(c+1)+...
Bessel equation x^2y"+xy'+(x^2-v^2)y=0
Coefficient recursion for Bessel equation___. (2m+2v)2ma(2m)+a(2m-2) =0 odd coefficients are zero
Bessel function Jn(x) is obtained by keeping v=n and a0=1/2^n n! then Bessel function for order n is___. x^nsum(m=0-inf.)(-1)^m x^2m/2^(2m+n)m!(n+m)!
Bessel function Jn(x)~ where ~ is asymptotically read sqrt(2/pi*x)cos(x-(n*pi/2)-(pi/4))
Gamma(v)=___ integral(0-inf)exp(-t)t^v-1dt
Jv(x)=___. x^vsum(m=0-inf)(-1)^mx^2m/2^(2m+v)m!gamma(v+m+1)
General solution of Bessel equation c1Jv(x)+c2J-v(x)if v is not a integer
If v is a integer then Jn(x) and J-n(x) are linearly dependent by___. J-n(x)=(-1)^nJn(x)
Derivative properties for bessel function (x^vJv(x))'=x^vJv-1(x) (x^-vJv(x))'=-x^-vJv+1(x)
Recurrence relations for Bessel function Jv-1(x)+Jv+1(x)=2vJv(x)/x Jv-1(x)-Jv+1(x)=2J'v(x)
J1/2(x)=___,J-1/2(x)=___. sqrt(2/pi*x)sinx sqrt(2/pi*x)cosx
If y"+py'+q=0 is substituted y=uv with v=exp(-integral(p/2dx)) then equation got changed to__if again substitute y=ux^-.5 then it is reduced to___. u"+[q-p^2/4-p'/2]u=0 x^2u"+(x^2+1/4-v^2)u=0
Second kind of Bessel function 2Jn(x)(lnx/2+k)/pi+x^nsum(m=0-inf)(-1)^m-1(hm + hm+n)x^2m/2^2m+nm!(m+n)!)/pi-x^-nsum(m=0-n-1)(n-m-1)!x^2m/2^2m-n m!)/pi
Yv(x)=___ (Jvcos(vpi)-J-v(x))/sin(vpi)
Sturm liouville problems [py']'+[q+kr]y=0
If any function is written as sum(m=0-inf)a(m)y(m)) then a(m) (f,ym)/(ym,ym)
||Pm(x)||= sqrt(2/2m+1)
Bessel inequality___. Parseval equality___. Sum(m=0-k)a(m)^2=<||f||^2 Sum(m=0-inf)a(m)^2=||f||^2
Hermite polynomials (-1)^nexp(x^2/2)d^n(exp(-x^2/2))/dx^n
Generating functions G=sum(m=0-inf)an(x)t^n where Hen=n!an(x) exp(tx-t^2/2)
Hen'(x)=___. nHen-1(x)
Hen(x) satisfies equation w=exp(-x^2/4)y satisfies Weber equations___. y''-xy'+ny=0 w''+(n+1/2-x^2/4)w=0
Integral transform___. F(s)=integral(0-inf)k(s,t)f(t)dt
Laplace equation___. Integral transform k(s,t)=exp(-st)
Linearity of Laplace transform___. L(af+bg)=aL(f)+bL(g)
Application of linearity of Laplace transform L(cosh(at))=___,L(sinh(at))=___. s/s^2-a^2 a/s^2-a^2
3Methods for deriving Laplace transform of cos,sin By calculus,by transforms using derivatives,by complex methods
L(cos(at))=___. L(sin(at))=___. s/s^2+a^2 a/s^2+a^2
L(t^a)=___where a is positive. Gamma(a+1)/s^a+1
L(exp(at)cos(wt))=___,L(exp(at)sin(wt))=___. s-a/(s-a)^2+w^2 w/(s-a)^2+w^2
If f(t) has a Laplace transform F(s) L(exp(at)f(t)) F(s-a)
What is existence theorem for Laplace transform? If f is defined and piecewise continuous on every finite interval on t=>0 and satisfies |f(t))<=Mexp(kt).This sufficient but not necessary.
Laplace transform L(f(^n))=___. s^nL(f)-s^n-1f(0)-s^n-2f'(0)-s^n-3f''(0)-...-f(^n-1)(0)
Let F(s) denote the transform of a function f(t) which is piecewise continuous for t>=0 and satisfies a growth restriction L(0integraltf(c)dc)=F(s)/s
L(u(t-a))=___. L(f(t-a)u(t-a))=___. exp(-as)/s exp(-as)F(s)
fk(t-a)=1/k a<=t<=a+k 0 otherwise Dirac delta function=___. lim(k->0)fk(t-a)
The Laplace transform of a piecewise continuous function f(t) with period p is L(f)=___. 0integralp(exp(-st)f(t)dt)/(1-exp(-ps))
Convolution of f and g___. H=0integraltf(c)g(t-c)dc
F'(s)=___. sintegral inf(F(s~)ds~)=___. L[tf(t)] L{f(t)/t)}
Laguerre's ODE___. Ln(t)=___. Recursion of Laguerre Polynomial ty''+(1-t)y'+ny=0 exp(t)dn(t^nexp(-t))/n!dtn (n+1)l(n+1)=(2n+1-t)l(n)-nl(n-1)
Generating function of Laguerre Polynomials exp(tx/(x-1))/(1-x)
For systems of ODEs y'=Ay+g therefore (A-sI)Y=-y(0)-G
Created by: jatint