matlab\specfun- Специализированные математические функции

Specialized math functions.

<airy> - Airy functions.

AIRY Airy functions.

W = AIRY(Z) is the Airy function, Ai(z), of the elements of Z.

W = AIRY(0,Z) is the same as AIRY(Z).

W = AIRY(1,Z) is the derivative, Ai'(z).

W = AIRY(2,Z) is the Airy function of the second kind, Bi(z).

W = AIRY(3,Z) is the derivative, Bi'(z).

If the argument Z is an array, the result is the same size.

[W,IERR] = AIRY(K,Z) also returns an array of error flags.

ierr = 1 Illegal arguments.

ierr = 2 Overflow. Return Inf.

ierr = 3 Some loss of accuracy in argument reduction.

ierr = 4 Complete loss of accuracy, z too large.

ierr = 5 No convergence. Return NaN.

The relationship between the Airy and modified Bessel functions is:

Ai(z) = 1/pi*sqrt(z/3)*K_1/3(zeta)

Bi(z) = sqrt(z/3)*(I_-1/3(zeta)+I_1/3(zeta))

where zeta = 2/3*z^(3/2)

AIRY uses a MEX interface to a Fortran library by D. E. Amos.

See also besselj, bessely, besseli, besselk.

Reference page in Help browser

doc airy

<besselj> - Bessel function of the first kind.

BESSELJ Bessel function of the first kind.

J = BESSELJ(NU,Z) is the Bessel function of the first kind, J_nu(Z).

The order NU need not be an integer, but must be real.

The argument Z can be complex. The result is real where Z is positive.

If NU and Z are arrays of the same size, the result is also that size.

If either input is a scalar, it is expanded to the other input's size.

If one input is a row vector and the other is a column vector, the

result is a two-dimensional table of function values.

J = BESSELJ(NU,Z,0) is the same as BESSELJ(NU,Z).

J = BESSELJ(NU,Z,1) scales J_nu(z) by exp(-abs(imag(z)))

[J,IERR] = BESSELJ(NU,Z) also returns an array of error flags.

ierr = 1 Illegal arguments.

ierr = 2 Overflow. Return Inf.

ierr = 3 Some loss of accuracy in argument reduction.

ierr = 4 Complete loss of accuracy, z or nu too large.

ierr = 5 No convergence. Return NaN.

Examples:

besselj(3:9,(0:.2:10)') generates the entire table on page 398

of Abramowitz and Stegun, Handbook of Mathematical Functions.

BESSELJ uses a MEX interface to a Fortran library by D. E. Amos.

Class support for inputs NU and Z:

float: double, single

See also bessely, besseli, besselk, besselh.

Reference page in Help browser

doc besselj

<bessely> - Bessel function of the second kind.

BESSELY Bessel function of the second kind.

Y = BESSELY(NU,Z) is the Bessel function of the second kind, Y_nu(Z).

The order NU need not be an integer, but must be real.

The argument Z can be complex. The result is real where Z is positive.

If nu and z are arrays of the same size, the result is also that size.

If either input is a scalar, it is expanded to the other input's size.

If one input is a row vector and the other is a column vector, the

result is a two-dimensional table of function values.

Y = BESSELY(NU,Z,0) is the same as BESSELY(NU,Z).

Y = BESSELY(NU,Z,1) scales Y_nu(z) by exp(-abs(imag(z)))

[Y,IERR] = BESSELY(NU,Z) also returns an array of error flags.

ierr = 1 Illegal arguments.

ierr = 2 Overflow. Return Inf.

ierr = 3 Some loss of accuracy in argument reduction.

ierr = 4 Complete loss of accuracy, z or nu too large.

ierr = 5 No convergence. Return NaN.

Examples:

bessely(3:9,(0:.2:10)') generates the entire table on page 399

of Abramowitz and Stegun, Handbook of Mathematical Functions.

BESSELY uses a MEX interface to a Fortran library by D. E. Amos.

Class support for inputs NU and Z:

float: double, single

See also besselj, besseli, besselk, besselh.

Reference page in Help browser

doc bessely

<besselh> - Bessel functions of the third kind (Hankel function).

BESSELH Bessel function of the third kind (Hankel function).

H = BESSELH(NU,K,Z), for K = 1 or 2, computes the Hankel function

H1_nu(Z) or H2_nu(Z) for each element of the complex array Z.

H = BESSELH(NU,Z) uses K = 1.

H = BESSELH(NU,K,Z,0) is the same as BESSELH(NU,K,Z)

H = BESSELH(NU,1,Z,1) scales H1_nu(z) by exp(-i*z)))

H = BESSELH(NU,2,Z,1) scales H2_nu(z) by exp(+i*z)))

If nu and z are arrays of the same size, the result is also that size.

If either input is a scalar, it is expanded to the other input's size.

If one input is a row vector and the other is a column vector, the

result is a two-dimensional table of function values.

[H,IERR] = BESSELH(NU,K,Z) also returns an array of error flags.

ierr = 1 Illegal arguments.

ierr = 2 Overflow. Return Inf.

ierr = 3 Some loss of accuracy in argument reduction.

ierr = 4 Complete loss of accuracy, z or nu too large.

ierr = 5 No convergence. Return NaN.

The relationship between the Hankel and Bessel functions is:

besselh(nu,1,z) = besselj(nu,z) + i*bessely(nu,z)

besselh(nu,2,z) = besselj(nu,z) - i*bessely(nu,z)

Example:

This example generates the contour plot of the modulus and

phase of the Hankel Function H1_0(z) shown on page 359 of

Abramowitz and Stegun, "Handbook of Mathematical Functions."

[X,Y] = meshgrid(-4:0.025:2,-1.5:0.025:1.5);

H = besselh(0,1,X+i*Y);

contour(X,Y,abs(H),0:0.2:3.2), hold on

contour(X,Y,(180/pi)*angle(H),-180:10:180); hold off

BESSELH uses a MEX interface to a Fortran library by D. E. Amos.

Class support for inputs NU and Z:

float: double, single

See also besselj, bessely, besseli, besselk.

Reference page in Help browser

doc besselh

<besseli> - Modified Bessel function of the first kind.

BESSELI Modified Bessel function of the first kind.

I = BESSELI(NU,Z) is the modified Bessel function of the first kind,

I_nu(Z). The order NU need not be an integer, but must be real.

The argument Z can be complex. The result is real where Z is positive.