# 15.3. Analyzing real-valued functions

*This is one of the 100+ free recipes of the IPython Cookbook, Second Edition, by Cyrille Rossant, a guide to numerical computing and data science in the Jupyter Notebook. The ebook and printed book are available for purchase at Packt Publishing.*

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SymPy contains a rich calculus toolbox to analyze real-valued functions: limits, power series, derivatives, integrals, Fourier transforms, and so on. In this recipe, we will show the very basics of these capabilities.

## How to do it...

**1. ** Let's define a few symbols and a function (which is just an expression depending on `x`

):

```
from sympy import *
init_printing()
```

```
var('x z')
```

```
f = 1 / (1 + x**2)
```

**2. ** Let's evaluate this function at 1:

```
f.subs(x, 1)
```

**3. ** We can compute the derivative of this function:

```
diff(f, x)
```

**4. ** What is \(f\)'s limit to infinity? (Note the double o (`oo`

) for the infinity symbol):

```
limit(f, x, oo)
```

**5. ** Here's how to compute a Taylor series (here, around 0, of order 9). The **Big O** can be removed with the `removeO()`

method.

```
series(f, x0=0, n=9)
```

**6. ** We can compute definite integrals (here, over the entire real line):

```
integrate(f, (x, -oo, oo))
```

**7. ** SymPy can also compute indefinite integrals:

```
integrate(f, x)
```

**8. ** Finally, let's compute \(f\)'s Fourier transforms:

```
fourier_transform(f, x, z)
```

## There's more...

SymPy includes a large number of other integral transforms besides the Fourier transform (http://docs.sympy.org/latest/modules/integrals/integrals.html). However, SymPy will not always be able to find closed-form solutions.

Here are a few general references about real analysis and calculus:

- Real analysis on Wikipedia, at https://en.wikipedia.org/wiki/Real_analysis#Bibliography
- Calculus on Wikibooks, at http://en.wikibooks.org/wiki/Calculus
- Real analysis on Awesome Math, at https://github.com/rossant/awesome-math/#real-analysis