Busy. Please wait.

show password
Forgot Password?

Don't have an account?  Sign up 

Username is available taken
show password


Make sure to remember your password. If you forget it there is no way for StudyStack to send you a reset link. You would need to create a new account.

By signing up, I agree to StudyStack's Terms of Service and Privacy Policy.

Already a StudyStack user? Log In

Reset Password
Enter the associated with your account, and we'll email you a link to reset your password.

Remove ads
Don't know
remaining cards
To flip the current card, click it or press the Spacebar key.  To move the current card to one of the three colored boxes, click on the box.  You may also press the UP ARROW key to move the card to the "Know" box, the DOWN ARROW key to move the card to the "Don't know" box, or the RIGHT ARROW key to move the card to the Remaining box.  You may also click on the card displayed in any of the three boxes to bring that card back to the center.

Pass complete!

"Know" box contains:
Time elapsed:
restart all cards

Embed Code - If you would like this activity on your web page, copy the script below and paste it into your web page.

  Normal Size     Small Size show me how


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