The function shifts each alphabetical character in the input string by a certain number of positions in the alphabet while maintaining the case and leaving digits unchanged.

[CODE]: 
```python
def approximate_function(input_str):
    result = ""
    shift = 17

    for char in input_str:
        if char.isalpha():
            start = 65 if char.isupper() else 97 
            result += chr(((ord(char) - start + shift) % 26) + start)
        else:
            result += char

    return result
```

Let me test some more inputs to further confirm this.

COMMAND: PYTHON(./temp/function.py "Hello" "WorLD" "1984xYz")