Kreyszing
Quiz yourself by thinking what should be in
each of the black spaces below before clicking
on it to display the answer.
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| Gauss hypergeometric equation | x(1-x)y"+[c-(a+b+1)x]y'-aby=0
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| Gauss hypergeometric equation solution | 1+abx/1!c+a(a+1)b(b+1)x^2/2!c(c+1)+...
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| Bessel equation | x^2y"+xy'+(x^2-v^2)y=0
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| Coefficient recursion for Bessel equation___. | (2m+2v)2ma(2m)+a(2m-2)
=0 odd coefficients are zero
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| 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)!
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| Bessel function Jn(x)~ where ~ is asymptotically read | sqrt(2/pi*x)cos(x-(n*pi/2)-(pi/4))
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| Gamma(v)=___ | integral(0-inf)exp(-t)t^v-1dt
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| Jv(x)=___. | x^vsum(m=0-inf)(-1)^mx^2m/2^(2m+v)m!gamma(v+m+1)
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| General solution of Bessel equation | c1Jv(x)+c2J-v(x)if v is not a integer
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| If v is a integer then Jn(x) and J-n(x) are linearly dependent by___. | J-n(x)=(-1)^nJn(x)
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| Derivative properties for bessel function | (x^vJv(x))'=x^vJv-1(x)
(x^-vJv(x))'=-x^-vJv+1(x)
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| Recurrence relations for Bessel function | Jv-1(x)+Jv+1(x)=2vJv(x)/x
Jv-1(x)-Jv+1(x)=2J'v(x)
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| J1/2(x)=___,J-1/2(x)=___. | sqrt(2/pi*x)sinx
sqrt(2/pi*x)cosx
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| 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
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| 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
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| Yv(x)=___ | (Jvcos(vpi)-J-v(x))/sin(vpi)
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| Sturm liouville problems | [py']'+[q+kr]y=0
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| If any function is written as sum(m=0-inf)a(m)y(m)) then a(m) | (f,ym)/(ym,ym)
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| ||Pm(x)||= | sqrt(2/2m+1)
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| Bessel inequality___. Parseval equality___. | Sum(m=0-k)a(m)^2=<||f||^2
Sum(m=0-inf)a(m)^2=||f||^2
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| Hermite polynomials | (-1)^nexp(x^2/2)d^n(exp(-x^2/2))/dx^n
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| Generating functions G=sum(m=0-inf)an(x)t^n where Hen=n!an(x) | exp(tx-t^2/2)
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| Hen'(x)=___. | nHen-1(x)
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| 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
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| Integral transform___. | F(s)=integral(0-inf)k(s,t)f(t)dt
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| Laplace equation___. | Integral transform k(s,t)=exp(-st)
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| Linearity of Laplace transform___. | L(af+bg)=aL(f)+bL(g)
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| Application of linearity of Laplace transform L(cosh(at))=___,L(sinh(at))=___. | s/s^2-a^2
a/s^2-a^2
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| 3Methods for deriving Laplace transform of cos,sin | By calculus,by transforms using derivatives,by complex methods
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| L(cos(at))=___. L(sin(at))=___. | s/s^2+a^2
a/s^2+a^2
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| L(t^a)=___where a is positive. | Gamma(a+1)/s^a+1
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| L(exp(at)cos(wt))=___,L(exp(at)sin(wt))=___. | s-a/(s-a)^2+w^2
w/(s-a)^2+w^2
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| If f(t) has a Laplace transform F(s) L(exp(at)f(t)) | F(s-a)
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| 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.
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| Laplace transform L(f(^n))=___. | s^nL(f)-s^n-1f(0)-s^n-2f'(0)-s^n-3f''(0)-...-f(^n-1)(0)
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| 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
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| L(u(t-a))=___. L(f(t-a)u(t-a))=___. | exp(-as)/s
exp(-as)F(s)
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| fk(t-a)=1/k a<=t<=a+k 0 otherwise Dirac delta function=___. | lim(k->0)fk(t-a)
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| 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))
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| Convolution of f and g___. | H=0integraltf(c)g(t-c)dc
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| L(H) | FG
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| F'(s)=___. sintegral inf(F(s~)ds~)=___. | L[tf(t)]
L{f(t)/t)}
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| 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)
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| Generating function of Laguerre Polynomials | exp(tx/(x-1))/(1-x)
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| For systems of ODEs y'=Ay+g therefore | (A-sI)Y=-y(0)-G
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Created by:
jatint
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