Allow specifying / calling functions with positional parameters

[cdp330]

Currently, calling functions seems to work like you are substituting symbols in an expression, e.g.

If $$y_{2}=x^{2}$$, then to get $$y(3)$$ we need to explictly type

$$y(x=3)$$

While this is convenient and intuitive for some expressions, it becomes annoying for others, because I have to specify the arguments at the call site every time.

This becomes especially inconvenient when integrating or summing user-defined functions, e.g.

If $$P_{X}=p\cdot\left(1-p\right)^{k-1}$$

then I can't write

$$\sum_{k=1}^7 \left( P_{X} \left( k \right) \right) $$

I have to write

$$\sum_{i=1}^7 \left( P_{X} \left( k = i \right) \right)$$

This is both annoying to type and hard to read, because of the temporary $$i$$ variable. What's more, as sheets become complex, it becomes harder to come up with unique names for temporary variables like $$i$$.

I propose a way to specify positional arguments for functions. This should work for both univariate and multivariable functions. For example, we could define $$length(x,y)=\sqrt{x^2 + y^2}$$, and call it via both

$$length (3,4)$$ or $$length(x=3, y=4)$$.

It would also be nice to be able to define multiple functions that use the same parameter names. For example, alongside with $$length(x,y)$$, you should be able to define $$length3D(x,y,z)$$.

Defining functions like this with the new Python Extensions feature is already possible, I would also like to be able to do it with functions written in normal notation.

[cdp330]

Because user-defined functions are essentially expressions at the moment, there is also the issue of re-using variables in functions. Currently if I have a "function" (really just expression) e.g. $$f=x^3$$, then I cannot define $$x=3$$ on another line, because then $$f$$ will evaluate to $$3^3$$ and no longer be "callable".

[cdp330]

I'm not sure how Mathcad handles these, but in SMath, a Mathcad clone, I can define two functions that both take in a parameter $$x$$, as well as a variable $$x$$ that is set to $$4$$. I can then evaluate either function with $$x$$ as an argument.

[mgreminger]

A live version of this response is available at https://engineeringpaper.xyz/sBWQfkiRRZnJxajjiN9QFG.

Integration, summation, and product variables work like a function substitution, so function notation is not needed. The following will work as expected:

$P_{X}=p\cdot\left(1-p\right)^{k-1}$

$\sum_{k=1}^7\left(P_{X}\right)= p^{7} - 7 \cdot p^{6} + 21 \cdot p^{5} - 35 \cdot p^{4} + 35 \cdot p^{3} - 21 \cdot p^{2} + 7 \cdot p$

The variables of integration, differentiation, summation, and integration are treated as local variables (similar to a user function call) and don't interfere with the rest of the sheet so they can be repeated as many times as necessary:

$\sum_{k=1}^5\left(k\right)= 15$

Functions can also share the same variable names without any issues:

$length=\sqrt{x^2+y^2}$

$length3D=\sqrt{x^2+y^2+z^2}$

$length\left(x=3,y=4\right)= 5$

$length3D\left(x=6,y=2,z=3\right)= 7$

[cdp330]

I was not aware that the variables of integration / summation etc. are treated as local variables, that is useful.

Nevertheless, if I define a function like $length=\sqrt{x^2 + y^2}$, this forbids me from using the $x$ as a global variable.

[mgreminger]

Yes, that's correct. User function notation allows variables to have multiple values. Otherwise, variables can only have one value. User functions were added for this purpose, to allow the local assignment of variables for the evaluation of an expression.