Notation issue with the integration of exponents.

[Fallen Seraph]

I'll not go into the details of the full question, because they are irrelevant to my problem.
Basically I have to integrate
$$\int_{0}^{\infty} exp (\iota\omega-\alpha)t dt$$

Which is a nice and easy integration, but it's putting in the limits that bothers me.

I simply wrote the exponent as $$((\iota \omega - \alpha)t)$$ because I didn't feel like writing an extra minus sign. I see no reason why I could not have written it

$$(-( \alpha -\iota \omega )t)$$

Which gives a finite answer when putting in the limits, whereas the first way of writing it gives an infinite answer. Could someone explain why one of these notations are incorrect?

 

[Shooting Star]

Integral[0, inf] exp(at)dt converges, i.e., has a finite value only when a<0.

 

[ogb p]

I understand your concern about notation and its potential impact on the accuracy of calculations. In this case, both notations are mathematically correct and should yield the same result. The only difference is the placement of the minus sign in the exponent, which does not affect the value of the integral.

However, it is important to maintain consistency in notation to avoid confusion and potential errors in calculations. In general, it is recommended to follow established conventions and use parentheses to clearly indicate the order of operations in mathematical expressions. This can help prevent confusion and ensure accurate results.

In this specific case, it is possible that the infinite answer obtained with the first notation may be due to a mistake in the integration process rather than the notation itself. I would suggest double-checking your calculations to ensure accuracy.

Overall, as long as the mathematical operations are correctly performed, the specific notation used should not affect the outcome of the integration.