Overload List
| # | Signature | Description |
|---|---|---|
| 1 | function Besj(NU: Double; Z: TCplx): TCplx; | Bessel function of the first kind, complex argument: J_(nu)(z). |
| 2 | procedure Besj(NU: Double; Z: TCplx; resCY: TVec; Scale: Boolean); | Sequence of Bessel functions of the first kind, complex argument: J_(nu+i)(z), i=0... N-1. |
| 3 | function Besj(NU: Double; Z: TCplx; Scale: Boolean): TCplx; | Bessel function of the first kind, complex argument, optional scaling: J_(nu)(z). |
| 4 | function Besj(NU: Double; A: Double): TCplx; | Bessel function of the first kind, real argument: J_(nu)(x). |
| 5 | function Besj(NU: Double; A: Double; Scale: Boolean): TCplx; | Bessel function of the first kind, real argument, optional scaling: J_(nu)(x). |
| 6 | function Besj(NU: Single; Z: TSCplx): TSCplx; | |
| 7 | function Besj(NU: Single; Z: TSCplx; Scale: Boolean): TSCplx; | |
| 8 | function Besj(NU: Single; A: Single): TSCplx; | |
| 9 | function Besj(NU: Single; A: Single; Scale: Boolean): TSCplx; |
Overload 1: function Besj(NU: Double; Z: TCplx): TCplx;
Bessel function of the first kind, complex argument: J_(nu)(z).
| # | Name | Description |
|---|---|---|
| 1 | Z | Defines complex value for which Jn(Z) should be evaluated. |
| 2 | NU | Defines the order of the Bessel function of the first kind Jn. |
Returns: TCplx (complex) - J_(nu)(z), the Bessel function of the first kind of order NU, evaluated at the complex argument Z.
J_(nu) is the solution of Bessel's equation z^2 w'' + z w' + (z^2-nu^2)w = 0 that is regular at the origin:
J_(nu)(z) = sum_(m=0)^(inf)((-1)^m)/(m! Gamma(m+nu+1))(z/2)^(2m+nu)
For negative orders the reflection formula J_(-nu)(z)=cos(nupi)J_(nu)(z)-sin(nupi)Y_(nu)(z) is applied. Domain. Any finite complex Z; order NU may be any real value.
Overload 2: procedure Besj(NU: Double; Z: TCplx; resCY: TVec; Scale: Boolean);
Sequence of Bessel functions of the first kind, complex argument: J_(nu+i)(z), i=0... N-1.
| # | Name | Description |
|---|---|---|
| 1 | NU | Defines the order of the Bessel function of the first kind Jn. |
| 2 | Z | Defines complex value for which Jn(Z) should be evaluated. |
| 3 | resCY | Contains the result, which is an array: resCY[I]=Exp(-Abs(Y))*J(FNU+I,Z) , I = 0...,N-1 , Y=Imag(Z). The part Exp(-Abs(Y)) is applied only if Scaling is True. The length of the array N is specified with resCY.Length on input. |
| 4 | Scale | If Scale parameter is true, Jn(Z) is multiplied by e^(-|Im z|). |
Result: stored in self (calling object)
Returns: Fills with J_(nu+i)(z) for i=0,...,N-1, where N is the input length of . When is true each element is multiplied by e^(-|Im z|).
resCY[i] = e^(-|Im z|) J_(nu+i)(z), i=0,...,N-1, the factor e^(-|Im z|) applied only when scaling is enabled. Domain. resCY must be a non-empty complex double vector; z=0 yields -inf.
Overload 3: function Besj(NU: Double; Z: TCplx; Scale: Boolean): TCplx;
Bessel function of the first kind, complex argument, optional scaling: J_(nu)(z).
| # | Name | Description |
|---|---|---|
| 1 | NU | Defines the order of the Bessel function of the first kind Jn. |
| 2 | Z | Defines complex value for which Jn(Z) should be evaluated. |
| 3 | Scale | If Scale parameter is true, Jn(Z) is multiplied by e^(-|Im z|). |
Returns: TCplx (complex) - J_(nu)(z) of order NU. When is true the result is multiplied by e^(-|Im z|), i.e. e^(-|Im z|) J_(nu)(z), which removes the exponential growth along the imaginary axis.
J_(nu)(z) = sum_(m=0)^(inf)((-1)^m)/(m! Gamma(m+nu+1))(z/2)^(2m+nu)
Domain. Any finite complex Z; order NU any real value. Behavior. On argument-reduction
failure for very large |z| the routine raises an exception.
Overload 4: function Besj(NU: Double; A: Double): TCplx;
Bessel function of the first kind, real argument: J_(nu)(x).
| # | Name | Description |
|---|---|---|
| 1 | NU | Defines the order of the Bessel function of the first kind Jn. |
| 2 | A | Defines real value for which Jn(A) should be evaluated. |
Returns: TCplx (complex) - J_(nu)(x) of order NU at the real argument A. The value is returned as a TCplx; for nu >= 0 and real x >= 0 the imaginary part is zero.
Same definition as the complex overload, J_(nu)(x)=sum_(m >= 0)((-1)^m)/(m! Gamma(m+nu+1))(x/2)^(2m+nu). Domain. Any finite real A; order NU any real value (negative orders use the reflection formula).
Overload 5: function Besj(NU: Double; A: Double; Scale: Boolean): TCplx;
Bessel function of the first kind, real argument, optional scaling: J_(nu)(x).
| # | Name | Description |
|---|---|---|
| 1 | NU | Defines the order of the Bessel function of the first kind Jn. |
| 2 | A | Defines real value for which Jn(A) should be evaluated. |
| 3 | Scale | If Scale parameter is true, Jn(A) is multiplied by e^(-|Im x|). |
Returns: TCplx (complex) - J_(nu)(x) of order NU. When is true the result is multiplied by e^(-|Im x|) (which equals 1 for real x, so scaling has no effect on a real argument).
Domain. Any finite real A; order NU any real value.
Overload 6: function Besj(NU: Single; Z: TSCplx): TSCplx;
| # | Name | Type | Description |
|---|---|---|---|
| 1 | NU | Single | scalar |
| 2 | Z | TSCplx | scalar |
Returns: TSCplx
Overload 7: function Besj(NU: Single; Z: TSCplx; Scale: Boolean): TSCplx;
| # | Name | Type | Description |
|---|---|---|---|
| 1 | NU | Single | scalar |
| 2 | Z | TSCplx | scalar |
| 3 | Scale | Boolean |
Returns: TSCplx
Overload 8: function Besj(NU: Single; A: Single): TSCplx;
| # | Name | Type | Description |
|---|---|---|---|
| 1 | NU | Single | scalar |
| 2 | A | Single | scalar |
Returns: TSCplx
Overload 9: function Besj(NU: Single; A: Single; Scale: Boolean): TSCplx;
| # | Name | Type | Description |
|---|---|---|---|
| 1 | NU | Single | scalar |
| 2 | A | Single | scalar |
| 3 | Scale | Boolean |
Returns: TSCplx