// International, 1989) or from the Cephes Mathematical Library, a
// commercial product. In either event, it is copyrighted by the author.
// What you see here may be used freely but it comes with no support or
// guarantee.
//
//   The two known misprints in the book are repaired here in the
// source listings for the gamma function and the incomplete beta
// integral.
//

// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
//
// Developed at SunPro, a Sun Microsystems, Inc. business.
// Permission to use, copy, modify, and distribute this
// software is freely granted, provided that this notice
// is preserved.
// ====================================================
//
//
// double log1p(double x)
//
// Method :
//   1. Argument Reduction: find k and f such that

// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
//
// Developed at SunPro, a Sun Microsystems, Inc. business.
// Permission to use, copy, modify, and distribute this
// software is freely granted, provided that this notice
// is preserved.
// ====================================================
//
// expm1(x)
// Returns exp(x)-1, the exponential of x minus 1.
//
// Method

// International, 1989) or from the Cephes Mathematical Library, a
// commercial product. In either event, it is copyrighted by the author.
// What you see here may be used freely but it comes with no support or
// guarantee.
//
//   The two known misprints in the book are repaired here in the
// source listings for the gamma function and the incomplete beta
// integral.
//

## Mix `Path`, `Query` and body parameters

First, of course, you can mix `Path`, `Query` and request body parameter declarations freely and **FastAPI** will know what to do.

And you can also declare body parameters as optional, by setting the default to `None`:

=== "Python 3.10+"

    ```Python hl_lines="18-20"

// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
//
// Developed at SunPro, a Sun Microsystems, Inc. business.
// Permission to use, copy, modify, and distribute this
// software is freely granted, provided that this notice
// is preserved.
// ====================================================
//
// __ieee754_jn(n, x), __ieee754_yn(n, x)
// floating point Bessel's function of the 1st and 2nd kind

    ```

=== "Python 3.8+"

    ```Python hl_lines="1  8"
    {!> ../../../docs_src/extra_models/tutorial005.py!}
    ```

## Recap

Use multiple Pydantic models and inherit freely for each case.

// International, 1989) or from the Cephes Mathematical Library, a
// commercial product. In either event, it is copyrighted by the author.
// What you see here may be used freely but it comes with no support or
// guarantee.
//
//   The two known misprints in the book are repaired here in the
// source listings for the gamma function and the incomplete beta
// integral.
//

     * declared in the given [KaSymbolWithMembers].
     *
     * It is worth noting that, while Java classes may contain declarations of static callables freely, in Kotlin only enum classes define
     * static callables. Hence, for non-enum Kotlin classes, it is not expected that the static declared member scope will contain any
     * callables.
     *

// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
//
// Developed at SunPro, a Sun Microsystems, Inc. business.
// Permission to use, copy, modify, and distribute this
// software is freely granted, provided that this notice
// is preserved.
// ====================================================
//
// __ieee754_lgamma_r(x, signgamp)
// Reentrant version of the logarithm of the Gamma function