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[✓] Compile incomplete gamma function with C code?

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Can I compile the Mathmatica incomplete gamma function Gamma(a,z), which accepts complex number with C code?

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The Compiled Function that can be compiled see list below. Probably Yes, Gamma function is on the list.

Compile`CompilerFunctions[] // Sort

{Abs, AddTo, And, Append, AppendTo, Apply, ArcCos, ArcCosh, ArcCot, \
ArcCoth, ArcCsc, ArcCsch, ArcSec, ArcSech, ArcSin, ArcSinh, ArcTan, \
ArcTanh, Arg, Array, ArrayDepth, Internal`Bag, Internal`BagPart, \
BitAnd, BitNot, BitOr, BitXor, Block, BlockRandom, Boole, Break, \
Cases, Catch, Ceiling, Chop, Internal`CompileError, \
System`Private`CompileSymbol, Complement, ComposeList, \
CompoundExpression, Conjugate, ConjugateTranspose, Continue, Cos, \
Cosh, Cot, Coth, Count, Csc, Csch, Decrement, Delete, DeleteCases, \
Dimensions, Divide, DivideBy, Do, Dot, Drop, Equal, Erf, Erfc, EvenQ, \
Exp, Internal`Expm1, Fibonacci, First, FixedPoint, FixedPointList, \
Flatten, NDSolve`FEM`FlattenAll, Floor, Fold, FoldList, For, \
FractionalPart, FreeQ, Gamma, Compile`GetElement, Goto, Greater, \
GreaterEqual, Gudermannian, Haversine, If, Im, Implies, Increment, \
Indexed, Inequality, Compile`InnerDo, Insert, IntegerDigits, \
IntegerPart, Intersection, InverseGudermannian, InverseHaversine, \
Compile`IteratorCount, Join, Label, Last, Length, Less, LessEqual, \
List, Log, Log10, Internal`Log1p, Log2, LogGamma, LogisticSigmoid, \
LucasL, Map, MapAll, MapAt, MapIndexed, MapThread, \
NDSolve`FEM`MapThreadDot, MatrixQ, Max, MemberQ, Min, Minus, Mod, \
Compile`Mod1, Module, Most, N, Negative, Nest, NestList, NonNegative, \
Not, OddQ, Or, OrderedQ, Out, Outer, Part, Partition, Piecewise, \
Plus, Position, Positive, Power, PreDecrement, PreIncrement, Prepend, \
PrependTo, Product, Quotient, Ramp, Random, RandomChoice, \
RandomComplex, RandomInteger, RandomReal, RandomSample, \
RandomVariate, Range, Re, RealAbs, RealSign, Internal`ReciprocalSqrt, \
ReplacePart, Rest, Return, Reverse, RotateLeft, RotateRight, Round, \
RuleCondition, SameQ, Scan, Sec, Sech, SeedRandom, Select, Set, \
SetDelayed, Compile`SetIterate, Sign, Sin, Sinc, Sinh, Sort, Sqrt, \
Internal`Square, Internal`StuffBag, Subtract, SubtractFrom, Sum, \
Switch, Table, Take, Tan, Tanh, TensorRank, Throw, Times, TimesBy, \
Tr, Transpose, Unequal, Union, Unitize, UnitStep, UnsameQ, VectorQ, \
Which, While, With, Xor}{Abs, AddTo, And, Append, AppendTo, Apply, ArcCos, ArcCosh, ArcCot, \
ArcCoth, ArcCsc, ArcCsch, ArcSec, ArcSech, ArcSin, ArcSinh, ArcTan, \
ArcTanh, Arg, Array, ArrayDepth, Internal`Bag, Internal`BagPart, \
BitAnd, BitNot, BitOr, BitXor, Block, BlockRandom, Boole, Break, \
Cases, Catch, Ceiling, Chop, Internal`CompileError, \
System`Private`CompileSymbol, Complement, ComposeList, \
CompoundExpression, Conjugate, ConjugateTranspose, Continue, Cos, \
Cosh, Cot, Coth, Count, Csc, Csch, Decrement, Delete, DeleteCases, \
Dimensions, Divide, DivideBy, Do, Dot, Drop, Equal, Erf, Erfc, EvenQ, \
Exp, Internal`Expm1, Fibonacci, First, FixedPoint, FixedPointList, \
Flatten, NDSolve`FEM`FlattenAll, Floor, Fold, FoldList, For, \
FractionalPart, FreeQ, Gamma, Compile`GetElement, Goto, Greater, \
GreaterEqual, Gudermannian, Haversine, If, Im, Implies, Increment, \
Indexed, Inequality, Compile`InnerDo, Insert, IntegerDigits, \
IntegerPart, Intersection, InverseGudermannian, InverseHaversine, \
Compile`IteratorCount, Join, Label, Last, Length, Less, LessEqual, \
List, Log, Log10, Internal`Log1p, Log2, LogGamma, LogisticSigmoid, \
LucasL, Map, MapAll, MapAt, MapIndexed, MapThread, \
NDSolve`FEM`MapThreadDot, MatrixQ, Max, MemberQ, Min, Minus, Mod, \
Compile`Mod1, Module, Most, N, Negative, Nest, NestList, NonNegative, \
Not, OddQ, Or, OrderedQ, Out, Outer, Part, Partition, Piecewise, \
Plus, Position, Positive, Power, PreDecrement, PreIncrement, Prepend, \
PrependTo, Product, Quotient, Ramp, Random, RandomChoice, \
RandomComplex, RandomInteger, RandomReal, RandomSample, \
RandomVariate, Range, Re, RealAbs, RealSign, Internal`ReciprocalSqrt, \
ReplacePart, Rest, Return, Reverse, RotateLeft, RotateRight, Round, \
RuleCondition, SameQ, Scan, Sec, Sech, SeedRandom, Select, Set, \
SetDelayed, Compile`SetIterate, Sign, Sin, Sinc, Sinh, Sort, Sqrt, \
Internal`Square, Internal`StuffBag, Subtract, SubtractFrom, Sum, \
Switch, Table, Take, Tan, Tanh, TensorRank, Throw, Times, TimesBy, \
Tr, Transpose, Unequal, Union, Unitize, UnitStep, UnsameQ, VectorQ, \
Which, While, With, Xor}

No, not possible for 2-argument Gamma.

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