Given the recurrence relation for the Legendre polynomials (2n + 1) xPn(x) = (n + 1) Pn + 1(x) + Pn - 1(x), which of the following integrals has a non-zero value?
A. \[\int_{ - 1}^{ + 1} {{x^2}{P_n}\left( x \right){P_{n + 1}}\left( x \right)dx} \]
B. \[\int_{ - 1}^{ + 1} {x{P_n}\left( x \right){P_{n + 2}}\left( x \right)dx} \]
C. \[\int_{ - 1}^{ + 1} {x{{\left[ {{P_n}\left( x \right)} \right]}^2}dx} \]
D. \[\int_{ - 1}^{ + 1} {{x^2}{P_n}\left( x \right){P_{n + 2}}\left( x \right)dx} \]
Answer: Option A
A. $$\frac{{1 + i}}{{\sqrt 2 }}a{\text{ and}} - \frac{{1 + i}}{{\sqrt 2 }}a$$
B. $$ia{\text{ and }} - ia$$
C. $$ia,\, - ia,\,\frac{{1 - i}}{{\sqrt 2 }}a{\text{ and}} - \frac{{1 - i}}{{\sqrt 2 }}a$$
D. $$\frac{{1 + i}}{{\sqrt 2 }}a,\, - \frac{{1 + i}}{{\sqrt 2 }}a,\,\frac{{1 - i}}{{\sqrt 2 }}a{\text{ and}} - \frac{{1 - i}}{{\sqrt 2 }}a$$
Which of the following functions of the complex variable z is not analytic everywhere?
A. ez
B. $$\sin \frac{{\text{z}}}{{\text{z}}}$$
C. e3
D. |z|3
A. \[\left( {1 - \sqrt 3 } \right){{{\bf{\hat i}}}^{\bf{'}}} + 3{{{\bf{\hat j}}}^{\bf{'}}} + \left( {1 + \sqrt 3 } \right){{{\bf{\hat k}}}^{\bf{'}}}\]
B. \[\left( {1 + \sqrt 3 } \right){{{\bf{\hat i}}}^{\bf{'}}} + 3{{{\bf{\hat j}}}^{\bf{'}}} + \left( {1 - \sqrt 3 } \right){{{\bf{\hat k}}}^{\bf{'}}}\]
C. \[\left( {1 - \sqrt 3 } \right){{{\bf{\hat i}}}^{\bf{'}}} + \left( {3 + \sqrt 3 } \right){{{\bf{\hat j}}}^{\bf{'}}} + 2{{{\bf{\hat k}}}^{\bf{'}}}\]
D. \[\left( {1 - \sqrt 3 } \right){{{\bf{\hat i}}}^{\bf{'}}} + \left( {3 - \sqrt 3 } \right){{{\bf{\hat j}}}^{\bf{'}}} + 2{{{\bf{\hat k}}}^{\bf{'}}}\]
Join The Discussion