This old post from dr Lopez is just what i am looking for 

Here is a example where @acer asked for:

Its like this by doing it by handwriting  
example : differential equation 

y" -2y' -3y=6 

Or for partial deratives in this notation f_xx   

I go further with this in Maple 

Note: i know my English is not that good, but i  like to write manual and not with deepl translation online

@Mariusz Iwaniuk

 Thanks

There is no doubt about it, that it has to do with the functional equation from Zeta(z)  , because i can always alter (convert) from maple input to 2d input 
But seeing the functional equation in Maple input is no problem either in your code.

By substituing : z=2+3*I (x>0, so in the rightplane) in the rightside of the functional equation you can calculate the leftside of the functional equation
(-z is in the left plane of the complex plane)
Now you extend the whole complex plane ,except for a pole z= 1) 

I start this thread about how to calculate the two function values from Zeta (z) for the two complexnumber  in the functional equation.
How precise for a decimal number here?

@Carl Love 
It was too much expected from me that the sum ( zeta function) could produce the Eulerproduct formula
The Zeta function as we now is not a average function
I try to understand more the importance for number theory of the zeta function.

@Mariusz Iwaniuk
 Thanks

It revolves between the connection  -z  and  z+1  
Its not directly clear not yet for me where your example stands for? 

@vv 

Thanks

You can say that the sum and eulerproduct( derived from the sum ) are equal after their proves. (only for real numbers then)

In general form both are only a notation then.
 

@vv 
Thanks

With this correct sum now Maple knows perhaps the Euler prorduct formula as closed formula?

@tomleslie 

Thanks

Your definition of the sum and product seems to be correct: they are equal , if look to the comparison
I am unsure if i defined this correct in my worksheet too
Is there a way in Maple doing this direct without filling in first some values in the sum and product?

@vv 

Thanks 
informative the wiki 

At least then : = 

Proving for all numbers in Maple is impossible , so how i can check  this symbolically ?
In order to check these sum and product if they are equal then i must fill in some values then. 
note: i think i must chance p in the sum into s  ( is real number >1 , or the same as  Re s>1) after looked in a book and seen in the @tom leslie code

@tomleslie 
Thanks

It is not that important for examing the graph of the zeta function i think?

@Carl Love 

Thanks

Those definitions can  be tricky

f = u +i v form   f = u(x,y) + i(x,y)  form see example 2.1

Question : use of `  hereunder ?

`f(x,y) ` = f(x,y); ` `;

About the one-argument and two argument form of arctan : for real arctan versus complex arctan ?

Download reaction_functionadvisor_arctan.mw

@The function 

I don't think it's a shame that the book does not come up with such answers you got here on the Maple Primes forum.
Here some people on the MaplePrimes  forum are really specialists in the Maple language.

@janhardo 

The FunctionAdvisor in Maple  gives two definitions for arctan

There are two approaches to defining a complex function in Maple.

Method 1. Make  f(x, y)  a function of two real variables  x, y :  f(x,y) 

Method 2. Make  f(z)  a function of the complex variable  z .  :  F(x+ I y)

@Carl Love 

Depends of the functionality of database what forum software can offer on search posts seems to me.
There is some "out the box"  forum software to install if i should start with making one
online forum on a server.
This forum has some base forum progam running what has added extended functionality further inbuilt ?

@Carl Love 

Thanks

In the list of all my question i like to filter out on who had answered my question 

Example: i search for your answers on my  questions 

@Carl Love 

Thanks

I got a answer from Maple , but it was not in a "bookform" , so i did it manually deriving the last answer from Maple to the wanted "bookform" 
Checking this manually derived answer in 2d maple form,  shows now it was the same as the answer in Maple for the derivation of the complex  integral.  
So, its all now solved for the derivation of the complex integral.