Calculus: Analytic Geometry and Calculus, with Vectors

174

Show that if (1) holds, then

(2)

(3)

(4)

Functions, limits, derivatives

Hn(x)e ax2l2 = (-1)n E ax2f2

dx

[-axH,(x) + H'(x)le axe/2+ = (-1)n dxn+i aax2l2

do+1

o+

axHn(x) - H(x) = (-1)n+iea==/ 2^12

d

'l dxn+ia -2

and

(5) axHn(x) - H.'a(x).

Use (5) and the fact that Ho(x) = 1 to obtain the formulas

Ho(x) = 1

Hi(x) = ax

H2(x) = a2x2 - a

Hs(x) = a°x° - 3a2x

H4(x) = a4x4 - 6a°x2 + 342

H5(x) = a6x6 - 10a4x° + 15asx

He(x) = a6x6 - 1Sa°x4 + 45a4x2 - 15a°

H7(x) = a7x7 - 21a6x° + 105a°x° - 105a4x.

23 The Laguerre polynomials are defined by the formulas Lo(x) = 1 and

L,,(x) = ex ...).

den

(xne x) (n = 1 ,2,3,

Show that

Lo(x) = 1

Ll(x) = -x + 1

L2(x) =x2-4x+2

Lo(x) = -x° + 9x2 - 18x + 6

L4(x) = x4 - 16x° + 72x2 - 96x + 24.

24 Supposing that y = eli6l or h(t) = eein °, use the chain rule and the formula

for derivatives of products to obtain the first three derivatives with respect to

t of these things. dns.:

(1) dd = h'(t) = eeia t cos t-

(2) d2y = h"(t) eein t sin t + e' ' COS2 t

(3) dt3 = h"'(t) _ -e'in' cos i - 3eBin' cos t sin t + e8in= cos° t.

25 Assuming existence of all of the derivatives we want to use, show that if

h(t) = f(g(t)), then

(1) Y(t) = f'(g(t))g'(t)

(2) h"(t) = f(g(t))g"(t) +f"(g(t))Ig'(t)l2

and write a formula for h"'(t). Then show that these formulas reduce to those

of Problem 24 when f(x) = el and g(t) = sin t.