Need some help with perfecting the layout

Question

\documentclass{article}
\usepackage{amsmath}
\setlength{\parindent}{0pt}
\newcommand{\fac}{n!}           
\newcommand\test[3]{\left(\begin{array}{@{\,}c@{\,}} #1 \\ #2 \end{array}\middle|\ \ #3 \right)}
\begin{document}

Dit gedeelte is gewijd aan de wiskundige berekening van de inproducten tussen de vectoren $u_0(E), u_1(E), v(x)$ en $w(p)$. Twee bestaande formules zullen hierbij van cruciaal belang blijken te zijn.\
De eerste formule zegt dat

\begin{align*}
\sum_{n=0}^{\infty} \frac{(a)_n}{\fac}F_1 \test{-n,b}{a}{y}_1 F_1& \test{-n}{a}{x} t^n\\
&=(1-t)^{b-a}(1-t+yt)^{-b} e^{\frac{xt}{t-1}}F_1 \test{b}{a}{\frac{xyt}{(1-t)(1-t+yt)}},\\
\end{align*}
waarbij $|t| <1.$\
De tweede formule is de volgende benadering:

\begin{equation*}
\lim_{z \rightarrow \infty} F_1 \test{c}{a}{z} = \lim_{z \rightarrow \infty} \frac{\Gamma(a)}{\Gamma(c)} e^z z^{c-a} \sum_{n=0}^{\infty} \frac{(a-c)_n (1-c)_n}{\fac} z^{-n}
\end{equation*}
\end{document}

---

Looking at the arrows you can see that the spacing is different. On the second picture the spacing between the letters seems bigger, the spacing between the two lines is also bigger

In the equation I didn't manage to get the alignment right. I'm assuming that one line is flushedleft and the other centered. The fraction in the first image is much smaller than in the 2nd one.

Answer 1

The first example is probably typeset like this, although I have made some slight changes that I will illustrate later.

\documentclass{article}
\usepackage[dutch]{babel}
\usepackage{amsmath}

%\setlength{\parindent}{0pt}
%\setlength{\parskip}{\medskipamount}

\newcommand{\fac}{n!}           
\newcommand\test[3]{%
  \left(\begin{array}{@{\,}c@{\,}}
  #1 \\ #2 \end{array}\middle|\;#3 \right)}

\begin{document}

Dit gedeelte is gewijd aan de wiskundige berekening van de inproducten 
tussen de vectoren $u_0(E)$, $u_1(E)$, $v(x)$~en~$w(p)$. Twee bestaande 
formules zullen hierbij van cruciaal belang blijken te zijn.

De eerste formule zegt dat
\begin{multline*}
\sum_{n=0}^{\infty} \frac{(a)_n}{\fac}F_1 \test{-n,b}{a}{y}_1 F_1 \test{-n}{a}{x} t^n\\
=(1-t)^{b-a}(1-t+yt)^{-b} e^{\frac{xt}{t-1}}F_1 \test{b}{a}{\frac{xyt}{(1-t)(1-t+yt)}},
\end{multline*}
waarbij $|t| <1$.

De tweede formule is de volgende benadering:
\begin{equation*}
\lim_{z \rightarrow \infty} F_1 \test{c}{a}{z} = 
\lim_{z \rightarrow \infty} \frac{\Gamma(a)}{\Gamma(c)} e^z z^{c-a} 
  \sum_{n=0}^{\infty} \frac{(a-c)_n (1-c)_n}{\fac} z^{-n}
\end{equation*}
\end{document}

1. The first formula is made with multline*.

2. Paragraphs don't end with \\, but with an empty line.

3. No empty line must precede a display.

4. No \\ must be typed at the end of a display.

The original document probably uses zero indentation and a non zero parskip, which I don't like at all. Just uncomment the two lines in the preamble (or, better, use the parskip package).

I have also changed the definition of the macro for the hypergeometric function: the space after the vertical bar is excessive.

Answer 2

While not reproducing exactly your first image, here is my take:

\documentclass[10pt]{article}
\usepackage{amsmath}
\setlength{\parindent}{0pt}
\newcommand{\fac}{n!}
\newcommand\test[3]{\left(\begin{array}{@{\,}c@{\,}} #1 \\ #2 \end{array}\middle|\ \ #3                 \right)}
\begin{document}

Dit gedeelte is gewijd aan de wiskundige berekening van de inproducten tussen de vectoren $u_0(E), u_1(E), v(x)$ en $w(p)$. Twee bestaande formules zullen hierbij van cruciaal belang blijken te zijn.
\medskip
De eerste formule zegt dat
\begin{multline*}
\sum_{n=0}^{\infty} \frac{(a)_n}{\fac}F_1 \test{-n,b}{a}{y}_1 F_1 \test{-n}{a}{x} t^n\\
=(1-t)^{b-a}(1-t+yt)^{-b} e^{\frac{xt}{t-1}}F_1 \test{b}{a}{\textstyle\frac{xyt}{(1-t)(1-    t+yt)}},
\end{multline*}%
waarbij $|t| < 1$.

\medskip
De tweede formule is de volgende benadering:
\begin{equation*}
\lim_{z \rightarrow \infty} F_1 \test{c}{a}{z} = \lim_{z \rightarrow \infty}         \frac{\Gamma(a)}{\Gamma(c)} e^z z^{c-a} \sum_{n=0}^{\infty} \frac{(a-c)_n (1-c)_n}{\fac}     z^{-n}
\end{equation*}
\end{document}

To get rid of the vertical space, I removed your explicit line breaks, and inserted \medskips.

The layout of the first equation is turned to multline*, so the first line will be flushed left, while the last line will be flushed right.

The fraction is made small with \textstyle, although I am not convinced that this looks better.

Setting everything in 10pt Computer Modern brings it quite close to your first image.