exp function

(Shortest import: from brian2 import exp)

brian2.units.unitsafefunctions.exp(x[, out])

Calculate the exponential of all elements in the input array.


x : array_like

Input values.


out : ndarray

Output array, element-wise exponential of x.

See also

Calculate exp(x) - 1 for all elements in the array.
Calculate 2**x for all elements in the array.


The irrational number e is also known as Euler’s number. It is approximately 2.718281, and is the base of the natural logarithm, ln (this means that, if \(x = \ln y = \log_e y\), then \(e^x = y\). For real input, exp(x) is always positive.

For complex arguments, x = a + ib, we can write \(e^x = e^a e^{ib}\). The first term, \(e^a\), is already known (it is the real argument, described above). The second term, \(e^{ib}\), is \(\cos b + i \sin b\), a function with magnitude 1 and a periodic phase.


[R19]Wikipedia, “Exponential function”, http://en.wikipedia.org/wiki/Exponential_function
[R20]M. Abramovitz and I. A. Stegun, “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,” Dover, 1964, p. 69, http://www.math.sfu.ca/~cbm/aands/page_69.htm


Plot the magnitude and phase of exp(x) in the complex plane:

>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-2*np.pi, 2*np.pi, 100)
>>> xx = x + 1j * x[:, np.newaxis] # a + ib over complex plane
>>> out = np.exp(xx)
>>> plt.subplot(121)
>>> plt.imshow(np.abs(out),
...            extent=[-2*np.pi, 2*np.pi, -2*np.pi, 2*np.pi])
>>> plt.title('Magnitude of exp(x)')
>>> plt.subplot(122)
>>> plt.imshow(np.angle(out),
...            extent=[-2*np.pi, 2*np.pi, -2*np.pi, 2*np.pi])
>>> plt.title('Phase (angle) of exp(x)')
>>> plt.show()