Chapter 11: Q10.5P (page 552)

(a) Express E₁(x)as an incomplete $Γ$ function.
(b) Find the asymptotic series for E₁(x).

Short Answer

Expert verified

The required expression is, $E_{1} (x) = \int_{x}^{\infty} t^{- 1} e^{- t} d t = Γ (0, x)$ .
The required series is, $E_{1} (x) = \frac{e^{- x}}{x} [1 - \frac{1}{x} + \frac{2}{x^{2}} - \frac{6}{x^{3}} + . . .]$ .

Step by step solution

Given information

E₁(x) and gamma function is given.

Definition of a Gamma function

The Gamma Function is defined as

$Γ (p) = \int_{0}^{\infty} x^{p - 1} e^{- x} d x, p > 0$

Begin with the definition of a Gamma function

(a)

Write the Gamma function.

$Γ (p) = \int_{0}^{\infty} x^{p - 1} e^{- x} d x, p > 0$

Substitute p = 0.

$\begin{array}{rcl} Γ (0, x) & = & \int_{0}^{\infty} t^{- 1} e^{- x} d \\ = & E_{1} (x) \end{array}$

Write the expansion of an incomplete Gamma function.

(b)

Expand the incomplete Gamma function.

$Γ (p, x) = x^{p - 1} e^{- x} [1 + \frac{(p - 1)}{x} + \frac{(p - 1) (p - 2)}{x^{2}} + \frac{(p - 1) (p - 2) (p - 3)}{x^{3}} + . . .]$

Substitute p = 0.

$\begin{array}{rcl} Γ (0, x) & = & E_{1} (x) \\ = & \frac{e^{- x}}{x} [1 - \frac{1}{x} + \frac{2}{x^{2}} - \frac{6}{x^{3}} + . . .] \end{array}$

Hence, the asymptotic series is $\begin{array}{rcl} \frac{e^{- x}}{x} [1 - \frac{1}{x} + \frac{2}{x^{2}} - \frac{6}{x^{3}} + . . .] \end{array}$ .

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(a) Express E₁(x)as an incomplete $Γ$ function.
(b) Find the asymptotic series for E₁(x).

Short Answer

Step by step solution

Given information

Definition of a Gamma function

Begin with the definition of a Gamma function

Write the expansion of an incomplete Gamma function.

One App. One Place for Learning.

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(a) Express E1(x)as an incompleteΓfunction.(b) Find the asymptotic series for E1(x).

Short Answer

Step by step solution

Given information

Definition of a Gamma function

Begin with the definition of a Gamma function

Write the expansion of an incomplete Gamma function.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electrostatics

Waves Physics

Radiation

Astrophysics

Further Mechanics and Thermal Physics

Conservation of Energy and Momentum

Study anywhere. Anytime. Across all devices.

(a) Express E₁(x)as an incomplete $Γ$ function.
(b) Find the asymptotic series for E₁(x).