# std::tgamma, std::tgammaf, std::tgammal

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< cpp‎ | numeric‎ | math

Defined in header `<cmath>` |  |
float tgamma ( float arg ); float tgammaf( float arg ); | (1) | (since C++11)
double tgamma ( double arg ); | (2) | (since C++11)
long double tgamma ( long double arg ); long double tgammal( long double arg ); | (3) | (since C++11)
double tgamma ( IntegralType arg ); | (4) | (since C++11)

1-3) Computes the gamma function of `arg`.
4) A set of overloads or a function template accepting an argument of any integral type. Equivalent to 2) (the argument is cast to double).

### Parameters

arg | - | value of a floating-point or Integral type

### Return value

If no errors occur, the value of the gamma function of `arg`, that is \(\mathsf{\Gamma}(arg) = \int_0^\infty t^{arg-1} e^{-t} \mathsf{d}t\)∫∞
0 t arg-1
e-t dt, is returned

If a domain error occurs, an implementation-defined value (NaN where supported) is returned.

If a pole error occurs, `±HUGE_VAL`, `±HUGE_VALF`, or `±HUGE_VALL` is returned.

If a range error due to overflow occurs, `±HUGE_VAL`, `±HUGE_VALF`, or `±HUGE_VALL` is returned.

If a range error due to underflow occurs, the correct value (after rounding) is returned.

### Error handling

Errors are reported as specified in math_errhandling.

If `arg` is zero or is an integer less than zero, a pole error or a domain error may occur.

If the implementation supports IEEE floating-point arithmetic (IEC 60559),

- If the argument is ±0, ±∞ is returned and FE_DIVBYZERO is raised
- If the argument is a negative integer, NaN is returned and FE_INVALID is raised
- If the argument is -∞, NaN is returned and FE_INVALID is raised
- If the argument is +∞, +∞ is returned.
- If the argument is NaN, NaN is returned

### Notes

If `arg` is a natural number, std::tgamma(arg) is the factorial of `arg-1`. Many implementations calculate the exact integer-domain factorial if the argument is a sufficiently small integer.

For IEEE-compatible type double, overflow happens if `0 < x < 1/DBL_MAX` or if `x > 171.7`

POSIX requires that a pole error occurs if the argument is zero, but a domain error occurs when the argument is a negative integer. It also specifies that in future, domain errors may be replaced by pole errors for negative integer arguments (in which case the return value in those cases would change from NaN to ±∞).

There is a non-standard function named `gamma` in various implementations, but its definition is inconsistent. For example, glibc and 4.2BSD version of `gamma` executes `lgamma`, but 4.4BSD version of `gamma` executes `tgamma`.

### Example

Run this code

    #include <iostream>
    #include <cmath>
    #include <cerrno>
    #include <cstring>
    #include <cfenv>
    #pragma STDC FENV_ACCESS ON
    int main()
    {
        std::cout << "tgamma(10) = " << std::tgamma(10)
                  << ", 9! = " << 2*3*4*5*6*7*8*9 << '\n'
                  << "tgamma(0.5) = " << std::tgamma(0.5)
                  << ", sqrt(pi) = " << std::sqrt(std::acos(-1)) << '\n';
        // special values
        std::cout << "tgamma(1) = " << std::tgamma(1) << '\n'
                  << "tgamma(+Inf) = " << std::tgamma(INFINITY) << '\n';
        // error handling
        errno=0;
        std::feclearexcept(FE_ALL_EXCEPT);
        std::cout << "tgamma(-1) = " << std::tgamma(-1) << '\n';
        if (errno == EDOM)
            std::cout << "    errno == EDOM: " << std::strerror(errno) << '\n';
        if (std::fetestexcept(FE_INVALID))
            std::cout << "    FE_INVALID raised\n";
    }

Possible output:

    tgamma(10) = 362880, 9! = 362880
    tgamma(0.5) = 1.77245, sqrt(pi) = 1.77245
    tgamma(1) = 1
    tgamma(+Inf) = inf
    tgamma(-1) = nan
        errno == EDOM: Numerical argument out of domain
        FE_INVALID raised

### See also

lgammalgammaflgammal (C++11)(C++11)(C++11) | natural logarithm of the gamma function (function)
betabetafbetal (C++17)(C++17)(C++17) | beta function (function)

### External links

Weisstein, Eric W. "Gamma Function." From MathWorld--A Wolfram Web Resource.

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