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# GammaRegularized

 GammaRegularizedis the regularized incomplete gamma function .
• Mathematical function, suitable for both symbolic and numerical manipulation.
• In non-singular cases, .
• GammaRegularized is the generalized regularized incomplete gamma function, defined in non-singular cases as Gamma[a, z0, z1]/Gamma[a].
• For certain special arguments, GammaRegularized automatically evaluates to exact values.
Evaluate numerically:
Evaluate numerically:
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 Scope   (4)
Evaluate to high precision:
The precision of the output tracks the precision of the input:
Evaluate numerically for complex arguments:
TraditionalForm formatting:
Evaluate at integer and half-integer arguments:
Infinite arguments give symbolic results:
GammaRegularized threads element-wise over lists:
GammaRegularized can be applied to a power series:
Series expansion at infinity:
Give the result for an arbitrary symbolic direction:
Evaluate at integer and half-integer arguments:
Series expansion at a generic point:
 Applications   (4)
Plot of the real part of GammaRegularized over the complex plane:
CDF of the distribution:
Calculate PDF:
Plot the CDFs for various degrees of freedom:
CDF of the gamma distribution:
Calculate PDF:
Plot the CDFs for various parameters:
Fractional derivatives/integrals of the exponential function:
Check that this is the defining Liouville integral:
Fractional derivative/integral of integer orders:
Plot fractional derivative/integral:
Use FullSimplify to simplify regularized gamma functions:
Use FunctionExpand to express regularized gamma functions through ordinary gamma functions:
Solve a transcendental equation:
Numerically find a root of a transcendental equation:
Large arguments can give results too small to be computed explicitly:
Machine-number inputs can give high-precision results:
Gamma rather than GammaRegularized is usually generated in computations:
Regularized gamma functions are typically not generated by FullSimplify:
Nest GammaRegularized over the complex plane:
Plot GammaRegularized at infinity:
Riemann surface of the incomplete regularized gamma function:
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