Asymptotic development of Integral of $e^xx^r$

Question

Let $\alpha \in (0,1)$ and $\delta \in (0,1/2)$ be fixed, and consider the following integrals for each integer $j \geq 0$: $$I_j(u):= \frac{e^u}{u^{j+\alpha}} \int_{-u\delta}^0 e^t t^{j-1+\alpha}\left(1+\frac{t}{u}\right)^{-1}dt, \hspace{2mm} u>0$$ Show that for any integer $k \geq 0$ and for any integer $0 \leq j \leq k$, each of these integrals as an asymptotic development of the form $$I_j(u) = \frac{e^u}{u^{\alpha+j}} \left( d_{0,j} + \frac{d_{1,j} }{u} + \cdots + \frac{d_{k,j}}{u^{k}} + O\left(\frac{1}{u^{k+1}}\right) \right) \text{ as }u \rightarrow \infty$$

Attempts: For $1 \leq j \leq k$, I expanded the $(1+t/u)^{-1}$ as a geometric series upto $k$ terms and got $$I_j(u) = \frac{e^u}{u^{j+\alpha}} \left(\sum_{n=0}^k \frac{(-1)^n}{u^n} \int_{-u \delta}^0 e^t t^{j-1+\alpha+n} dt + O\left(\frac{1}{u^{k+1}}\right) \right)$$ So, I'm down to showing there exist constants $c_{0, j}, \cdots , c_{k, j}$ such that for each $0 \leq n \leq k$, $$\int_{-u \delta}^0 e^t t^{j-1+\alpha+n} dt = c_{n, j} + O \left( \frac{1}{u^{k+1}} \right)$$

But I'm not sure how to proceed from here (I tried integrating by parts but got too many terms...does that look promising?) or if this will work. Also will something similar work for $j=0$? Thanks.

P.S.: This is related to this unanswered question I asked on MSE a long time ago: https://math.stackexchange.com/questions/3667949/integrating-an-asymptotic-development

With notation there, what I had done was separate the integral of $\gamma$ into one over $\gamma_1$ and one over $\gamma_2$, and tried obtaining an asymptotic development of the desired form for $\gamma_1$ (with that along $\gamma_2$ being analogous). For $\gamma_1$, I substituted $x:=e^u$ (the original integral there is in terms of $x$) and with $x>1$, we have $u>0$. The integral so obtained is equal to the real integral $$\int_{1-\delta}^1 e^{uv} (v-1)^{j-1+\alpha} v^{-1} dv$$ wherein I made the substitution $v := 1 + t/u$ to get the above collection of integrals. Perhaps those integrals could be directly/better estimated by complex analytic methods, in which case I would also appreciate a direct estimation of the integral I started with in the link.

Answer 1

You want to show that for some complex $b$ we have $$\int_{-v}^0 e^t t^a\,dt=b+O(v^{-c})$$ where $v:=u\delta\to\infty$, $a:=j-1+\alpha+n$, $c:=k+1$.

This is true. Indeed, let $b:=\int_{-\infty}^0 e^t t^a\,dt$. Then, by l'Hospital's rule, for any real $c$ $$\Big|b-\int_{-v}^0 e^t t^a\,dt\Big|\le\int_{-\infty}^{-v} e^t |t|^a\,dt \sim e^{-v} v^a=O(v^{-c})$$ as $v\to\infty$.