Chapter 11 Special functions Mathematical methods in the physical sciences 3rd edition Mary L. Boas Chapter 11 Special functions Lecture 12 Gamma, beta, error, and elliptic

2. The factorial function (usually, n : integer)

3. Definition of the gamma function: recursion relation (p: noninteger) - Example

4. The Gamma function of negative numbers - Example - Using the above relation, 1) Gamma(p= negative integers)  infinite. 2) For p < 0, the sign changes alternatively in the intervals between negative integers

5. Some important formulas involving gamma functions

6. Beta functions

7. Beta functions in terms of gamma functions

- Example

8. The simple pendulum - Example 1 For small vibration,

- Example 2 In case of 180 swings (-90 to +90)

9. The error function (useful in probability theory) - Standard model or Gaussian cumulative distribution function - Complementary error function - in terms of the standard normal cumulative distribution function

- Several useful facts - Imaginary error function:

10. Asymptotic series

- This series diverges for every x because of the factors in the numerator. For large enough x, the higher terms are fairly small and then negligible. For this reason, the first few terms give a good approximation. (asymptotic series)

11. Stirling’s formula - Stirling’s formula

11. Elliptic integrals and functions - Legendre forms: - Jacobi forms:

- Complete Elliptic integrals (=/2, x=sin=1): - Example 1

- Example 2

- Example 4. Find arc length of an ellipse. (using computer or tables)

- Example 5. Pendulum swing through large angles.

- For =30, this pendulum would get exactly out of phase with one of very small amplitude in about 32 periods.

- Elliptic Functions