Mathematica: Why isn't this command with Assuming working?

[AxiomOfChoice]

Mathematica: Why isn't this command with "Assuming" working?

I'm trying to execute the following command:

Assuming[g >= 0 && t >= 0, Refine[Abs[1 + I g t ]]]

I expect it to spit out

$$\sqrt{1+t^2g^2},$$

but instead I'm just getting

$$\text{Abs}[1 + i g t],$$

which is obviously pretty worthless. Does anyone see what I'm doing wrong?

 

[gabbagabbahey]

I'd try using Simplify[] or FullSimplify[] instead of Refine[].

 

[Bill Simpson]

What is "simplest" is a deeply subjective issue.
Usually Mathematica uses the smallest LeafCount to decide that.

In[1]:= LeafCount[Abs[1-I a b]]
Out[1]= 9

In[2]:= LeafCount[Sqrt[1+a^2 b^2]]
Out[2]= 13

So Mathematica thinks your preferred output is more complicated. Trying to subvert what Mathematica thinks it wants to do is usually very difficult. It is possible to write custom functions to be used by LeafCount but I have never had any success doing that.

In[3]:= Refine[Abs[1-a b I],a>=0&&b>=0]
Out[3]= Abs[1-I a b]

In[4]:= Simplify[Abs[1-a b I],a>=0&&b>=0]
Out[4]= Abs[1-I a b]

In[5]:= FullSimplify[Abs[1-a b I],a>=0&&b>=0]
Out[5]= Abs[1-I a b]

So none of those are going to, by default, accomplish what you want.

However

In[6]:= Refine[Abs[1-a I],a>0]
Out[6]= Abs[1-I a]

In[7]:= Simplify[Abs[1-a I],a>0]
Out[7]= Abs[1-I a]

In[8]:= FullSimplify[Abs[1-a I],a>=0]
Out[8]= Sqrt[1 + a^2]

So FullSimplify can do it with a single variable and not with more.