Team Quanta gladly presents all short questions of Methods of Mathematical Physics – II’s Chapter#3: Bassel Function.
Q.1 what is definition of Bessel’s Function?
Answer: The standard form of boundary differential equation is;
$$x^2y^{\prime\prime}+xy^\prime+\left(x^2-y^2\right)=0$$
The solution of this equation is known as Bessel function.
Q.2 why Bessel function is used for?
Answer: Bessel’s function arises when finding separable solution to Laplace’s equation and the Helmholtz equation in cylindrical or spherical coordinates. Bessel functions are especially important for many problems of wave propagation and static potentials.
Q.3 where do Bessel function come from?
Answer: Bessel’s function also called cylindrical function any of a set of mathematical functions systematically derived around 1819 by the German astronomer Friedrich Wilhelm. Bessel during investigation of solutions of one of Keplar’s equations of planetary motion.
Q.4 Are Bessel function continuous?
Answer: It is the piecewise continuous function, generally the non-homogeneous term of the problem. This orthogonal series expansion is also known as a Fourier Bessel series expansion or a Generalized Fourier series expansion.
Q.5 what is the Bessel function if 1st kind?
Answer: Bessel function of 1st kind is $J_a$ the series indicates that $-J_a$ is the derivative of $J_a\left(x\right)$ much like $-\sin{x}$ is the derivative of cosx; more generally, the derivative of $J_a\left(x\right)$ can be expressed in terms of $J_{n\pm1}(x)$ by the identities.
Q.6 Are Bessel functions real?
Answer: If the argument is real and the order V is integer, the Bessel function is real.
Q.7 what is Bessel function of first and 2nd kind?
Answer: 1st kind is; $\ J_v\left(x\right)$ in the solution to Bessel’s equation is referred to as a Bessel function of the 1st kind.
2nd kind; $Y_v\left(x\right)$ in the solution to Bessel’s equation is referred to as a Bessel function of the 2nd kind or sometimes the Weber function or the Neumann function.
Q.8 Is Bessel function normalized?
Answer: All the Bessel functions have a useful normalization property and satisfy a differential equation.
Q.9 what is recurrence relation with example?
Answer: A recurrence relation is an equation that recursively defines a sequence where the next term is a function of the previous terms. For example; Fibonacci series
$$F_n=F_{n-1}+F_{n-2}$$
Tower of Hanoi; $F_n=2F_n-1+1$
Q.10 what is recurrence used for?
Answer: Recurrence relation is used to reduce complicated problems to an iterative process based on simple versions of the problems. An example problem in which this approach can be used is the Tower of Hanoi Puzzle.
Q.11 what is the degree of recurrence relation?
Answer: The degree of recurrence relation is ‘ ‘K’’ if the highest terms of the numeric function is expressed in the terms of its previous K term.
Q.12 what are the methods for solving recurrence relation?
Answer: The methods are;
- Substitution method
- Iteration method
- Recursion method
- Master method
Q.13 what are the two types of recurrence?
Answer: Type-I; Divide and conquer recurrence relation. Type-II; Linear recurrence relation. These types of recurrence relations can be easily solved by using substitution method.
Q.14 what is first order recurrence relation?
Answer: A recurrence relation of the form;
$$a_n=Ca_n+F\left(n\right)\ \ for\ n\geq1$$
Where C is a constant and F(n) is a unknown function and is called linear recurrence relation of 1st order with constant coefficient. If F(n)=0, the relation is homogeneous and otherwise its non-homogeneous.
Q.15 what is the 2nd order recurrence relation?
Answer: A 2nd order linear homogeneous recurrence relation with constant coefficients is a recurrence relation of the form;
$$ak=Aak-1+Bak-2\$$
For all integers k greater than some fixed integer, where A and B are fixed real numbers with B = 0
Q.16 what is the order of recurrence relation?
Answer: The order of the recurrence relation or difference equation is defined to be the difference between the highest and lowest subscripts of F(x) or $\ a_r=y_k$
Q.17 which is a method to solve recurrence?
Answer: The master method is a formula for solving recurrence relations of the form;
$$T_n=aT\left(nlb\right)+F\left(n\right)$$
Where; n is size of input and a is number of sub problems in the recursion. Nlb is size of each sub problem. All sub problems are assumed to have the same size.
Q.18 what is recurrence relation for an algorithm?
Answer: The recurrence relation is;
$$T\left(n\right)=3+T\left(n-1\right)+T\left(n-2\right)$$
To solve this, you would use the iterative method. Start expanding the terms until you find the pattern. For this example, you would expand T(n-1) to get;
$$T\left(n\right)=6+2\ast T\left(n-2\right)+T\left(n-3\right)$$
Then expand T(n-2) to get;
$$T\left(n\right)=12+3\ast T\left(c-3\right)+2\ast T\left(n-4\right)$$
Q.19 how many equivalence relations are possible in a set A={1,2,3}?
Answer: Only two possible relations are there which are equivalence.
Q.20 what recurrence means?
Answer: A new occurrence of something that happened or appeared before.
$$a_n=4^\ast2^n-1^\ast3^n={\frac{7}{2}}^\ast2^n-{\frac{1}{2}}^\ast3^n$$
Q.21 what is the recurrence relation for 12345?
Answer: Therefore the solution to the recurrence relation is;
Q22 what are the formulas of recurrence relation?
Answer: The formulas are;
$$H_n^\prime\left(x\right)=2nH_{n-1}\left(x\right)$$
$$H_{n+1}\left(x\right)=2xH_n\left(x\right)-2nH_{n-1}\left(x\right)$$
