Special function usage in Matlab (confluent hypergeometric function, Kummer function, Bessel function, etc.)

Matlab's support and description of special functions is not as flexible and detailed as mathematica, and its functions are also inferior, but it is basically enough for engineering students. Let me make some conclusions below.

Confluent hypergeometric function

See the Wikipedia entry
Confluent hypergeometric function

Confluent Hypergeometric Function of the First Kind

For the definition of this function, please refer to the special function-related textbooks (recommended Wang Zhuxi version or Liu Shishi version), or refer to the Wolfram link
https://mathworld.wolfram.com/ConfluentHypergeometricFunctionoftheFirstKind.html

The corresponding function of this function in Mathematica is Hypergeometric1F1[a,b,z], and the corresponding function in matlab is hypergeom(a,b,z).
Help link for this function in matlab https://www.mathworks.com/help/symbolic/hypergeom.html#bt1nkmw-2

This function in matlab is actually a generalized hypergeometric function, which means ${}_p{F}_q(a;b;z)$ , that is, a and b here can be vectors. The equivalent function in MMA is actuallyHypergeometricPFQ.

Confluent Hypergeometric Function of the Second Kind

The function definition can be found in the link:
https://mathworld.wolfram.com/ConfluentHypergeometricFunctionoftheSecondKind.html

The corresponding function of this function in Mathematica is HypergeometricU[a,b,z], and the corresponding function in matlab is kummerU(a,b,z).
This definition and help documentation in matlab can easily be confused with the confluent hypergeometric function of the first kind. Sometimes, however, this function is indeed referred to directly as the Kummer function. But in order to avoid confusion, it is best to use the first category and the second category in MMA.

Bessel function

Matlab does a good job with Bessel functions. The details are as follows, and the specific syntax can be found in the help documentation

Function name	meaning
besselj	Bessel function of the first kind
besseli	Modified Bessel function of the first kind
bessely	Bessel functions of the second kind
besselk	Modified Bessel function of the second kind
besselh	Bessel functions of the third kind (Hankel functions)