Whether a Bessel function can reduce to an elementary function and explain when it can be reduced to an elementary function Formula used: The Bessel’s function for n order is J n ( x ) = 2 π x cos ( x − n π 2 − π 4 ) . Calculation: When the Bessel function reduce to an elementary function, that Bessel’s function is called elementary Bessel’s function. The reduction of elementary Bessel’s function, from Bessel function is possible when the value of v is ± 1 2 , ± 3 2 , ± 5 2 , ⋅ ⋅ ⋅ . As an example of elementary Bessel’s function is calculated as: J 1 2 ( x ) = 2 π x cos ( x − 1 2 × π 2 − π 4 ) = 2 π x cos ( x − π 4 − π 4 ) = 2 π x cos ( x − 2 π 4 ) Further simplification of above equation. J 1 2 ( x ) = 2 π x cos ( x − π 2 ) = 2 π x cos ( π 2 − x ) = 2 π x sin ( x ) Where cos ( π 2 − x ) = sin x and cos ( − x ) = cos ( x ) .

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