Incomplete Gamma Function of the Second Kind
05-26-2021, 12:32 AM
Post: #2
 Albert Chan Senior Member Posts: 2,347 Joined: Jul 2018
RE: Incomplete Gamma Function of the Second Kind
(05-25-2021 10:26 PM)toml_12953 Wrote:  "This function works only in radian mode Error: Bad Argument Value"

It doesn't use any trig functions that I can see so why the error?

Depending on user input, CAS might not recognize it related to gamma function.
It may tried to remove infinity by substitution, x = tan(t), so that x→∞ map to t→pi/2

CAS> gamma1(a,x) := int(t^(a-1)*e^-t, t=x .. inf)

CAS> gamma1(1, i) ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → e^-i
CAS> gamma1(1.0, i) ﻿ ﻿ ﻿ → integrate(tan(t)^0.*e^(-tan(t))*(1+tan(t)^2), t, atan(i), π/2)

Actually, x=tan(t) here just shift the infinity ... atan(i)

Does atan(i) even have a limit ? CAS seems to think it is just a value.

CAS> limit(atan(x),x=i) ﻿ ﻿ ﻿ → ATAN(i)

Ref: DLMF: Chapter 8 Incomplete Gamma and Related Functions
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 Messages In This Thread Incomplete Gamma Function of the Second Kind - toml_12953 - 05-25-2021, 10:26 PM RE: Incomplete Gamma Function of the Second Kind - Albert Chan - 05-26-2021 12:32 AM