Question

the value of J'0 (x) is​

Answer 1

Answer:

Bessel functions

Explanation:

Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation

{\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+\left(x^{2}-\alpha ^{2}\right)y=0}{\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+\left(x^{2}-\alpha ^{2}\right)y=0}

for an arbitrary complex number α, the order of the Bessel function. Although α and −α produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of α.

The most important cases are when α is an integer or half-integer. Bessel functions for integer α are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer α are obtained when the Helmholtz equation is solved in spherical coordinates.

Answer 2

Answer:

Concept:

Bessel functions are the canonical solutions y(x) of Bessel's differential equation, which was first established by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel.

Explanation:

Bessel's differential equation has solutions known as Bessel functions of the first kind, abbreviated as J(x). When x is positive or integer, Bessel functions of the first kind are finite at the origin (x = 0), but when x is non-integer or negative, they diverge as x gets closer to zero. By using the Frobenius method on Bessel's equation, it is possible to define the function by its series expansion around x = 0:

                   ${\displaystyle J_{\alpha }(x)=\sum _{m=0}^{\infty }{\frac {(-1)^{m}}{m!\Gamma (m+\alpha +1)}}{\left({\frac {x}{2}}\right)}^{2m+\alpha }$

                 

where the factorial function is generalized to non-integer values by the shifted gamma function, or (z). If is an integer, the Bessel function of the first kind is a complete function; if not, it is a multivalued function with a singular value at zero.

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