The Bessel functions of the first kind are defined as the solutions to the Bessel differential equation
(1)

which are nonsingular at the origin. They are sometimes also called cylinder functions or cylindrical harmonics. The above plot shows for , 1, 2, ..., 5. The notation was first used by Hansen (1843) and subsequently by Schlömilch (1857) to denote what is now written (Watson 1966, p. 14). However, Hansen's definition of the function itself in terms of the generating function
(2)

is the same as the modern one (Watson 1966, p. 14). Bessel used the notation to denote what is now called the Bessel function of the first kind (Cajori 1993, vol. 2, p. 279).
The Bessel function can also be defined by the contour integral
(3)

where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416).
The Bessel function of the first kind is implemented in the Wolfram Language as BesselJ[nu, z].
To solve the differential equation, apply Frobenius method using a series solution of the form
(4)

Plugging into (1) yields
(5)

(6)

The indicial equation, obtained by setting , is
(7)

Since is defined as the first nonzero term, , so . Now, if ,
(8)

(9)

(10)

(11)

First, look at the special case , then (11) becomes
(12)

so
(13)

Now let , where , 2, ....
(14)
 
(15)
 
(16)

which, using the identity , gives
(17)

Similarly, letting ,
(18)

which, using the identity , gives
(19)

Plugging back into (◇) with gives
(20)
 
(21)
 
(22)
 
(23)
 
(24)

The Bessel functions of order are therefore defined as
(25)
 
(26)

so the general solution for is
(27)

Now, consider a general . Equation (◇) requires
(28)

(29)

for , 3, ..., so
(30)
 
(31)

for , 3, .... Let , where , 2, ..., then
(32)
 
(33)

where is the function of and obtained by iterating the recursion relationship down to . Now let , where , 2, ..., so
(34)
 
(35)
 
(36)

Plugging back into (◇),
(37)
 
(38)
 
(39)
 
(40)
 
(41)

Now define
(42)

where the factorials can be generalized to gamma functions for nonintegral . The above equation then becomes
(43)

Returning to equation (◇) and examining the case ,
(44)

However, the sign of is arbitrary, so the solutions must be the same for and . We are therefore free to replace with , so
(45)

and we obtain the same solutions as before, but with replaced by .
(46)

We can relate and (when is an integer) by writing
(47)

Now let . Then
(48)
 
(49)

But for , so the denominator is infinite and the terms on the left are zero. We therefore have
(50)
 
(51)

Note that the Bessel differential equation is secondorder, so there must be two linearly independent solutions. We have found both only for . For a general nonintegral order, the independent solutions are and . When is an integer, the general (real) solution is of the form
(52)

where is a Bessel function of the first kind, (a.k.a. ) is the Bessel function of the second kind (a.k.a. Neumann function or Weber function), and and are constants. Complex solutions are given by the Hankel functions (a.k.a. Bessel functions of the third kind).
The Bessel functions are orthogonal in according to
(53)

where is the th zero of and is the Kronecker delta (Arfken 1985, p. 592).
Except when is a negative integer,
(54)

where is the gamma function and is a Whittaker function.
In terms of a confluent hypergeometric function of the first kind, the Bessel function is written
(55)

A derivative identity for expressing higher order Bessel functions in terms of is
(56)

where is a Chebyshev polynomial of the first kind. Asymptotic forms for the Bessel functions are
(57)

for and
(58)

for (correcting the condition of Abramowitz and Stegun 1972, p. 364).
A derivative identity is
(59)

An integral identity is
(60)

Some sum identities are
(61)

(which follows from the generating function (◇) with ),
(62)

(Abramowitz and Stegun 1972, p. 363),
(63)

(Abramowitz and Stegun 1972, p. 361),
(64)

for (Abramowitz and Stegun 1972, p. 361),
(65)

(Abramowitz and Stegun 1972, p. 361), and the JacobiAnger expansion
(66)

which can also be written
(67)

The Bessel function addition theorem states
(68)

Various integrals can be expressed in terms of Bessel functions
(69)

which is Bessel's first integral,
(70)
 
(71)

for , 2, ...,
(72)

for , 2, ...,
(73)

for . The Bessel functions are normalized so that
(74)

for positive integral (and real) . Integrals involving include
(75)

(76)

Ratios of Bessel functions of the first kind have continued fraction
(77)

(Wall 1948, p. 349).
The special case of gives as the series
(78)

(Abramowitz and Stegun 1972, p. 360), or the integral
(79)
