Differential Operator Notation Question

**sael** · May 12th 2010, 04:19 PM

Hello,

I am very new to differential operators ( and operators in general ). I was doing fine with understanding them until I came across this notation:

$(De^{\gamma})(x)=\gamma e ^{\gamma x}$

Now I am thoroughly lost because the left hand side doesn't make any sense to me.

Any help very appreciated.

**drumist** · May 12th 2010, 04:42 PM

That notation is ambiguous which is why you don't generally see it very often, but I believe it is trying to convey this:

$\frac{d}{dx} \left( e^{\gamma x} \right) = \gamma e^{\gamma x}$

which of course is true for constant $\gamma$ .

Basically if we have a function $f$ defined by $f(x)=e^{\gamma x}$ , then we would say $Df$ is the derivative of $f$ (which is more commonly referred to as $f'$ or $\frac{df}{dx}$ ).

**sael** · May 12th 2010, 04:55 PM

Could $e^{\gamma}$ be treated like an operator so that the product $D e^{\gamma}$ is also an operator?

If that was true would that then allow me to write $(D e^{\gamma})(x)$ as $D((e^{\gamma})x)$ ?

**drumist** · May 12th 2010, 06:24 PM

Well, I am not exactly an expert on the notation. Like I said, it's usually only used in rare cases, and when it is used, it's usually clear what it means. The example you posted is somewhat unusual because, as far as I can tell, it's not even valid notation. I stated in my last post what I believe was intended by the notation, but I do not accept that the notation as written is valid. But again, I'd yield to someone who is more familiar with it and can explain why the notation is valid.

$D$ is an operator that maps a function to another function (its derivative). If I am given a function $f$ , then $Df$ is the derivative of $f$ . That is, $Df$ is itself a function. This isn't a "product" because we aren't multiplying $D$ with $f$ .

So you can evaluate that function at a value $x$ , i.e., $(Df)(x)$ . The extra parentheses are a bit confusing, but are necessary to clarify that we want to evaluate the function $Df$ at $x$ . If we wrote $Df(x)$ it may appear we are multiplying $f(x)$ by some number $D$ .

So, the left-hand side of the notation you posted really means evaluate the function $De^\gamma$ at $x$ . Which means $e^\gamma$ should be a function. The problem is that we don't really know what $e^\gamma$ means without guessing.

I'm being pedantic here, and I think this is probably a minor point in whatever you are working on. So, I might suggest ignoring what you posted, and instead consider this example instead:

Suppose $f$ is a function such that $f(x)=e^{\gamma x}$ . Then $(Df)(x) = \gamma e^{\gamma x}$ .

**sael** · May 12th 2010, 06:49 PM

Thank you, it does make it easier to think of the problem in the way you have described

Just for reference that notation appears in the book; Advanced Engineering Mathematics, 9th, Kreyszig, section 2.3.

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