Gain insights into Python for scientific computing. Explore arrays, plotting, linear equations, and algorithms using NumPy, Matplotlib, SciPy. Delve into applying tools with practical exercises.

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Python3

If you're a scientist or an engineer interested in learning scientific computing, this is the place to start.

In this course, you'll learn to write your own useful code to perform impactful scientific computations. Along the way, your understanding will be tested with periodic quizzes and exercises.

Topics covered in this course include arrays, plotting, linear equations, symbolic computation, scientific algorithms, and random variables. You’ll also be exposed to popular Python packages like NumPy, Matplotlib, SciPy, and others. In the last part of the course, the application section will test your ability to recall and apply the tools you have studied into newly learned scientific concepts.

At the end of this course, you'll be equipped with the tools necessary for everyday scientific computation.

Python for Scientists and Engineers

# Task
Sometimes the integrals of complex functions are difficult to compute and the result is not as *clean*. For example:

$$\int tan^{-1}(x) \space dx=x\space tan^{-1}(x) -\frac{1}{2}ln(1+x^2)+C$$

Integrals of complex functions are simplified by approximating integrals of a simplified function using Taylor polynomials. The Taylor series of $tan^{-1}(x)$ is given as a simple addition of polynomials.

$$tan^{-1}(x)=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\frac{x^9}{9}...$$

Let's apply this in Python as well.

## Problem statement

Define a **Python** function `ts_integral()` that computes the indefinite or definite integral of the Taylor series from the input mathematical function.

The function should have the following arguments in this order:

**Obligatory Arguments** - The function should always have these arguments at least. 
1. The mathematical function input: `f`.
2. The variable to be integrated: `x`.

**Optional Arguments** - the function will input defaults even if the user does not provide these.

3. The order of the Taylor series expansion `n`, with the default value set to `5`.
4. The limits of integration; `lim1` and `lim2`.

```python 3
def ts_integral(f, x, n, lim1, lim2) 
```

**Return Statement**

The function should return a tuple with two values: 

1. The Taylor series of the input function. 
2. The integral from the Taylor series of the input function. The value of the integral should be up to **3** significant figures.


In this exercise, you will implement a Python function to integrate complex mathematical functions.

Introduction

Python Refresher

Arrays

Plotting

Systems of Linear Equations

Symbolic Computation

Scientific Algorithms

Random Variables

Applications

Conclusion

Appendix

Python for Scientists and Engineers Exam

Exercise: Integrating Complex Functions

Task