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Gamma Function

Calculate ฮ“(x) and ln(ฮ“(x)) for real numbers. For positive integers, ฮ“(n) = (n-1)!. Uses Lanczos approximation.

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Why: Understanding gamma function helps you make better, data-driven decisions.

How: Enter x (real number), Decimal Places to calculate results.

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ฮ“
SPECIAL FUNCTIONSStatistics

Gamma Function ฮ“(x)

Compute ฮ“(x) and ln(ฮ“(x)). For positive integers, ฮ“(n) = (n-1)!. Lanczos approximation for general reals.

๐Ÿ“š Quick Examples โ€” Click to Load

Inputs

ฮ“(x) is undefined for x = 0, -1, -2, ...

gamma_result
CALCULATED
ฮ“(x)
ฮ“(1) = 1.000000
ln(ฮ“(x))
0.000000
Factorial form
ฮ“(1) = (1-1)! = 0!
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Gamma Function Calculator
ฮ“(1) = 1.000000
ln(ฮ“) = 0.0000
numbervibe.com/calculators/mathematics/statistics/gamma-function-calculator

ฮ“(x) for x โˆˆ (0, 5]

๐Ÿ“ Calculation Steps

INPUT
Input x
1
RESULT
ฮ“(x)
1.000000
ฮ“(1) = (1-1)!
ln(ฮ“(x))
0.000000
ext{Natural} \text{log} ext{of} ext{gamma}

For educational and informational purposes only. Verify with a qualified professional.

๐Ÿ“‹ Key Takeaways

  • โ€ข ฮ“(n) = (n-1)! for positive integers; extends factorial to real/complex numbers
  • โ€ข ฮ“(1/2) = โˆšฯ€; ฮ“(1) = ฮ“(2) = 1
  • โ€ข ฮ“(z) has poles at z = 0, -1, -2, ... (undefined)
  • โ€ข Used in probability (gamma, beta, chi-squared), combinatorics, analysis

๐Ÿ’ก Did You Know?

๐Ÿ“ฮ“(1/2) = โˆšฯ€ appears in the normal distribution and many integrals.Source: Probability
๐Ÿ”ขFor n โˆˆ โ„•, ฮ“(n+1) = n! โ€” the factorial is a special case.Source: Factorial
๐Ÿ“‰ฮ“(z) has simple poles at 0, -1, -2, ... with residues (-1)^n/n!Source: Complex Analysis
๐Ÿ“Šln(ฮ“(x)) is often used in computations to avoid overflow for large x.Source: Numerics

๐Ÿ“Š Infographic Stats

ฮ“(1)
= 1
ฮ“(1/2)
= โˆšฯ€ โ‰ˆ 1.772
ฮ“(5)
= 4! = 24
ฮ“(0)
undefined

๐Ÿ“– How It Works

For Re(z) > 0, ฮ“(z) = โˆซโ‚€^โˆž t^(z-1) e^(-t) dt. For positive integers, ฮ“(n) = (n-1)!. For other real z, we use the Lanczos approximation: a series that converges quickly and is accurate to many decimal places.

ฮ“(z+1) = zยทฮ“(z) (recurrence)

ฮ“(z)ยทฮ“(1-z) = ฯ€ / sin(ฯ€z) (reflection)

๐ŸŽฏ Expert Tips

Use ln(ฮ“) for large x

ฮ“(x) grows very fast. ln(ฮ“(x)) avoids overflow in probability computations.

Negative arguments

Use reflection: ฮ“(-z) = ฯ€/(sin(ฯ€z)ยทฮ“(1+z)). Undefined at 0, -1, -2, ...

๐Ÿ“‹ Comparison Table

xฮ“(x)
11
21
32
46
524
0.5โˆšฯ€ โ‰ˆ 1.772

โ“ FAQ

Why is ฮ“(n) = (n-1)! and not n!?

Historical convention. The recurrence ฮ“(z+1) = zยทฮ“(z) with ฮ“(1)=1 gives ฮ“(n)=1ยท2ยท...ยท(n-1)=(n-1)!.

Where is the gamma function used?

Probability (gamma, beta, chi-squared, Student t), combinatorics, number theory, physics, and many integrals.

What about ฮ“(0) and negative integers?

ฮ“ has simple poles there โ€” undefined. The reflection formula gives finite values for other negative reals.

โš ๏ธ Disclaimer: Results use Lanczos approximation. For very large or small x, numerical precision may vary. Not for critical applications without verification.

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