pracma (version 1.8.8)

einsteinF: Einstein Functions

Description

Einstein functions.

Usage

einsteinF(d, x)

Arguments

x

numeric or complex vector.

d

parameter to select one of the Einstein functions E1, E2, E3, E4.

Value

• Numeric/complex scalar or vector.

Details

The Einstein functions are sometimes used for the Planck-Einstein oscillator in one degree of freedom.

The functions are defined as: $$E1(x) = \frac{x^2 e^x}{(e^x - 1)^2}$$ $$E2(x) = \frac{x}{e^x - 1}$$ $$E3(x) = ln(1 - e^{-x})$$ $$E4(x) = \frac{x}{e^x - 1} - ln(1 - e^{-x})$$

E1 has an inflection point as x=2.34694130....

Examples

x1 <- seq(-4, 4, length.out = 101)
y1 <- einsteinF(1, x1)
plot(x1, y1, type = "l", col = "red",
             xlab = "", ylab = "", main = "Einstein Function E1(x)")
grid()

y2 <- einsteinF(2, x1)
plot(x1, y2, type = "l", col = "red",
             xlab = "", ylab = "", main = "Einstein Function E2(x)")
grid()

x3 <- seq(0, 5, length.out = 101)
y3 <- einsteinF(3, x3)
plot(x3, y3, type = "l", col = "red",
             xlab = "", ylab = "", main = "Einstein Function E3(x)")
grid()

y4 <- einsteinF(4, x3)
plot(x3, y4, type = "l", col = "red",
             xlab = "", ylab = "", main = "Einstein Function E4(x)")
grid()