11.7 Use the result of Exercise 6 to show by induction that each basis function \( S_{(n)} \) is symmetric; that is, \[ S_{(n)}(p+x)=S_{(n)}(p-x) \] for all \( x \), where \( p=\frac{1}{2}(n+1) h \).
Added by Katrina A.
Close
Step 1
We need to show that \( S_{(1)}(p+x) = S_{(1)}(p-x) \). Given \( p = \frac{1}{2}(1+1)h = h \), we need to verify that \( S_{(1)}(h+x) = S_{(1)}(h-x) \). Assume \( S_{(1)}(x) \) is symmetric by definition or from Exercise 6. Thus, the base case holds. Show more…
Show all steps
Your feedback will help us improve your experience
Carson Merrill and 52 other Calculus 1 / AB educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Show that $\sum_{0}^{n}(2 l+1) P_{1}(x)=P_{n}^{\prime}(x)+P_{n+1}^{\prime}(x) . H i n t:$ Use mathematical induction: (a) Verify the formula for $n=0$. (b) Assuming that it is true for $l=n-1$ show [using $(5.8 \mathrm{c})]$ that it is true for $l=n$.
SERIES SOLUTIONS OF DIFFERENTIAL EQUATIONS; LEGENDRE POLYNOMIALS; BESSEL FUNCTIONS; SETS OF ORTHOGONAL FUNCTIONS
Miscellaneous problems
The Hermite polynomials $H_{n}(x)$ are orthogonal on the interval $(-\infty, \infty)$ with respect to the weight function $W(x)=e^{-x^{2}}$ . Verify this fact for the first three Hermite polynomials: $$H_{0}(x) \equiv 1, \quad H_{1}(x)=2 x, \quad H_{2}(x)=4 x^{2}-2$$
Partial Differential Equations
Fourier Series
Show that the Legendre polynomials of even degree are even functions of $x,$ while those of odd degree are odd functions.
Series Solutions of Differential Equations
Special Functions
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Watch the video solution with this free unlock.
EMAIL
PASSWORD