Defining Functions#

import numpy as np

Mathematical Functions#

All the standard mathematical functions are available in NumPy:

Function

NumPy Syntax

\(\sin(x)\)

np.sin(x)

\(\cos(x)\)

np.cos(x)

\(\tan(x)\)

np.tan(x)

\(\tan^{-1}(x)\)

np.arctan(x)

\(e^x\)

np.exp(x)

\(\ln(x)\)

np.log(x)

\(\log_{10}(x)\)

np.log10(x)

\(\sqrt{x}\)

np.sqrt(x)

Let’s compute some examples:

np.cos(np.pi/4)

0.7071067811865476

1/np.sqrt(2)

0.7071067811865475

np.log(2)

0.6931471805599453

np.arctan(1)

0.7853981633974483

np.log10(2)

0.3010299956639812

10**0.3010299956639812

2.0

Defining Functions#

We can define our own custom functions. For example, let’s write a function called fun which takes input parameters x and y and returns the value

\[ \sqrt{x^2 + y^2} \]
def fun(x,y):
    value = np.sqrt(x**2 + y**2)
    return value

Let’s make some key observations about the construction:

  • def is a keyword that starts the function definition

  • fun is the name of the function

  • the input parameters are listed within parentheses (x,y)

  • def statement ends with a colon :

  • body of the function is indented 4 spaces

  • output value follows the return keyword

fun(1,2)

2.23606797749979

fun(-1,1)

1.4142135623730951

lambda Functions#

We can also create simple functions using the lambda keyword. For example, let’s create the function:

\[ f(x) = \frac{1}{1 + x^2} \]
f = lambda x: 1/(1 + x**2)

The keyword lambda starts the function definition, the input variable is x and the formula for the funciton output follows a colon :.

Compute some values of \(f(x)\):

f(0)

1.0

f(1)

0.5

f(2)

0.2

Now let’s construct the function

\[ h(x,y) = \sqrt{x^2 + y^2} \]
h = lambda x,y: np.sqrt(x**2 + y**2)

h(1,2)

2.23606797749979

h(-1,1)

1.4142135623730951

Lambda functions are helpful for defining simple functions but longer more complicated functions require the def construction.