How to define a mathematical function in SymPy?

Question

I've been trying this now for hours. I think I don't understand a basic concept, that's why I couldn't answer this question to myself so far.

What I'm trying is to implement a simple mathematical function, like this:

f(x) = x**2 + 1

After that I want to derive that function.

I've defined the symbol and function with:

x = sympy.Symbol('x')
f = sympy.Function('f')(x)

Now I'm struggling with defining the equation to this function f(x). Something like f.exp("x**2 + 1") is not working.

I also wonder how I could get a print out to the console of this function after it's finally defined.

Solution 1

sympy.Function is for undefined functions. Like if f = Function('f') then f(x) remains unevaluated in expressions.

If you want an actual function (like if you do f(1) it evaluates x**2 + 1 at x=1, you can use a Python function

def f(x):
    return x**2 + 1

Then f(Symbol('x')) will give a symbolic x**2 + 1 and f(1) will give 2.

Or you can assign the expression to a variable

f = x**2 + 1

and use that. If you want to substitute x for a value, use subs, like

f.subs(x, 1)

Solution 2

Here's your solution:

>>> import sympy
>>> x = sympy.symbols('x')
>>> f = x**2 + 1
>>> sympy.diff(f, x)
2*x

Solution 3

Another possibility (isympy command prompt):

>>> type(x)
<class 'sympy.core.symbol.Symbol'>
>>> f = Lambda(x, x**2)
>>> f
     2
x ↦ x 
>>> f(3)
9

Calculating the derivative works like that:

>>> g = Lambda(x, diff(f(x), x))
>>> g
x ↦ 2x
>>> g(3)
6

Comments

[Robin]

For what is sympy.Function uses, when not for defining functions?

[enedil]

I suppose it's a base class for a whole range of standard functions.

[asmeurer]

@Robin sympy.Function is for undefined functions. Like if f = Function('f') then f(x) remains unevaluated in expressions.

[Bill]

Thanks for this. Do you still need to do f = Function('f') before defining the function or is that implicit from the python function definition?

[asmeurer]

@Bill def f(...) sets the variable f to be the Python function defined by the def. If you do f = Function('f'), that will override the variable f to be the SymPy object Function('f'). The two options I show here do not use Function at all. That is only if you want something that is completely unevaluated. See also the section in the SymPy tutorial on symbols.

[sɐunıɔןɐqɐp]

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