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The ordinary integral fails because the simple real poles at $-2$ and $3$ produce nonintegrable logarithmic singularities. Partial fractions giveClosing above for $\omega >0$ and below for $\omega <0$, with symmetric semicircular indentations, yieldsTaking real and imaginary parts,All integrals here run over $\mathbb{R}$.

The normal linear model is $Y=X\beta +\epsilon$ with $\epsilon \sim N(0,{\sigma }^2{I}_n)$. Full column rank givesWith hat matrix $H=X(X^{T}X{)}^{-1}{X}^T$, the $i$th regression leverage is $h_{ii}$. Since $e=(I-H)Y=(I-H)\epsilon$ and $I-H$ is an orthogonal projection,High-leverage observations can move their own fitted values strongly and may exert disproportionate influence; residual size alone can therefore conceal them.

Boundary-layer theory assumes steady incompressible high-Reynolds-number flow, $\delta /x\ll 1$, weak streamwise viscous diffusion, nearly constant pressure across the layer, and outer pressure gradient $-{\rho }^{-1}{p}_x=U{U}^{\prime }$.

For $\varphi =-A{r}^k\cos (k\theta )/k$,At $\theta =\pm \pi /k$ the normal velocity vanishes; along the upper wall $U(x)=A{x}^{k-1}$. Boundary-layer scaling givesSet $m=k-1$. The Falkner-Skan equation becomesso $\alpha =k-1$, $\beta =-k/2$, $\gamma =1-k$, with
$f(0)=f^{\prime }(0)=0$ and $f^{\prime }(\infty )=1$. Finallyrequires $x\gg (\nu /A{)}^{1/k}$; thus $a=1/k$ and the tip region violates the approximation.