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Chapter 11: Q11.7P (page 554)

The function $蠄 (p) = \frac{d}{d p} I n 螕 (p)$ is called the digamma function, and the polygama functions are defined by $蠄^{n} (p) = \frac{d^{n}}{d p^{n}} 蠄 (p)$ . [Warning: Some authors define $蠄 (p)$ as $\frac{d}{d p} I n (p!) = \frac{d}{d p} I n 螕 (p + 1)$ ).]
(a) Show that $蠄 (p + 1) = 蠄 (p) + \frac{1}{p}$ . Hint: See (3.4).
(b) Use Problem 6 to obtain $蠄 (p) 鈻� I n (p) - \frac{1}{2 p} - \frac{1}{12 p^{2}} . . .$ .

Short Answer

Expert verified

The statements have been proved.

Step by step solution

01

Given Information

The function is $蠄 (p) = \frac{d}{d p} I n 螕 (p)$ .

02

Definition of the sterling’s formula.

Sterling鈥檚 formula is used to simplify formulas involving factorial.

$n! 鈻� n^{n} e^{- n} \sqrt{2 蟺苍}$ .

03

Prove the statement ψ(p+1)=ψ(p)+1p.

The function is $蠄 (p) = \frac{d}{d p} I n 螕 (p)$ .

$蠄 (p) = \frac{d}{d p} I n 螕 (p) 蠄 (p + 1) = \frac{d}{d p} I n 螕 (p + 1) 蠄 (p + 1) = \frac{d}{d p} I n 螕 (p) 蠄 (p + 1) = 蠄 (p) + \frac{1}{p}$

Hence, the statement has been proved.

04

Prove the statement ψ(p)□In(p)-12p-112p2.

The function is $蠄 (p) = \frac{d}{d p} I n 螕 (p)$

$I n 螕 (p) = n I n n - n + I n \sqrt{\frac{2 蟺}{n}} + I n (1 + \frac{1}{12 n} + . . .) .$

Substitute the value mentioned above in the equation mentioned below.

$蠄 (p) 鈻� \frac{d}{d p} I n 螕 (p) 蠄 (p) 鈻� I n n + 1 - 1 - \frac{1}{2 n} + \frac{d}{d n} (1 + \frac{1}{12 n} + . .) 蠄 (p) 鈻� I n n - \frac{1}{2 n} + \frac{d}{d n} (1 + \frac{1}{12 n} + . .)$

Hence, the statement has been proved.

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Most popular questions from this chapter

Computer plot graphs of

(a) E_n(x) for n = 0-10and x = 0.2.

(b) E₁(x) and E_n(x)for x = 0-2.

(c) the sine integral $S_{i} (x) = {鈭�}_{0}^{x} \frac{s i n t}{t} d t$ and the cosine integral $C_{i} (x) = {鈭�}_{0}^{鈭�} \frac{C o s t}{t} d t$ for $x = 0 - 4 蟺$ .

Expand the integrands of Kand E[see ( 12.3 )] in power series in $k^{2} s i n^{2} 胃$ (assuming small k ), and integrate term by term to find power series approximations for the complete elliptic integrals Kand E.

Express the complementary error function erfc(x)as an incomplete $螕'$ function (see Problem 2) and use your result in Problem 2to obtain (again) the asymptotic expansion oferfc(x) as in (10.4) .

Use the recursion relation (3.4), and if needed, equation (3.2) to simplify:

$螕 (\frac{2}{3}) / 螕 (\frac{8}{3})$

The integral (3.1) is improper because of infinite upper limit and it is also improper for 0 < p < 1 because x^p-1becomes infinite at the lower limit. However, the integral is convergent for any p>0. Prove this.

See all solutions

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