36 Integrals with Coalescing Saddles Properties 36.10 Differential Equations 36.12 Uniform Approximation of Integrals

§36.11 Leading-Order Asymptotics

Notes:: The formulas in this section are derived by the method of stationary phase, applied to the real critical points of the integral representations in §36.2. See §2.3(iv) and also Berry and Howls (1991). For (36.11.4) the integral is exponentially small when and the dominant contribution is from a critical point off the real axis.
Keywords:: Pearcey integral, canonical integrals, elliptic umbilic canonical integral, hyperbolic umbilic canonical integral, swallowtail canonical integral
Referenced by:: §36.15(ii)
Permalink:: http://dlmf.nist.gov/36.11

With real critical points (36.4.1) ordered so that

36.11.1 $t_{1}(\mathbf{x})<t_{2}(\mathbf{x})<\cdots<t_{{j_{{\max}}}}(\mathbf{x}),$

Symbols:: $t_{j}(\mathbf{x})$ : solutions
Permalink:: http://dlmf.nist.gov/36.11.E1
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and far from the bifurcation set, the cuspoid canonical integrals are approximated by

36.11.2 $\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)=\sqrt{2\pi}\sum\limits% _{{j=1}}^{{j_{{\max}}(\mathbf{x})}}\mathop{\exp\/}\nolimits\!\left(i\left(% \mathop{\Phi_{{K}}\/}\nolimits\!\left(t_{j}(\mathbf{x});\mathbf{x}\right)+% \tfrac{1}{4}\pi(-1)^{{j+K+1}}\right)\right)\left|\frac{{\partial}^{2}\mathop{% \Phi_{{K}}\/}\nolimits\!\left(t_{j}(\mathbf{x});\mathbf{x}\right)}{{\partial t% }^{2}}\right|^{{-1/2}}(1+\mathop{o\/}\nolimits\!\left(1\right)).$

Symbols:: $\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\Phi_{{K}}\/}\nolimits\!\left(t;\mathbf{x}\right)$ : cuspoid catastrophe, $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : variable, : codimension and $t_{j}(\mathbf{x})$ : solutions
Permalink:: http://dlmf.nist.gov/36.11.E2
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¶ Asymptotics along Symmetry Lines

36.11.3 $\mathop{\Psi_{{2}}\/}\nolimits\!\left(0,y\right)=\begin{cases}\sqrt{\ifrac{\pi% }{y}}\left(\mathop{\exp\/}\nolimits\!\left(\tfrac{1}{4}i\pi\right)+\mathop{o\/% }\nolimits\!\left(1\right)\right),&y\to+\infty,\\ \sqrt{\ifrac{\pi}{|y|}}\mathop{\exp\/}\nolimits\!\left(-\tfrac{1}{4}i\pi\right% )\left(1+i\sqrt{2}\mathop{\exp\/}\nolimits\!\left(-\frac{1}{4}iy^{2}\right)+% \mathop{o\/}\nolimits\!\left(1\right)\right),&y\to-\infty.\end{cases}$

Symbols:: $\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than and : real parameter
Permalink:: http://dlmf.nist.gov/36.11.E3
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36.11.4 $\mathop{\Psi_{{3}}\/}\nolimits\!\left(x,0,0\right)=\frac{\sqrt{2\pi}}{(5|x|^{3% })^{{1/8}}}\begin{cases}\mathop{\exp\/}\nolimits\!\left(-2\sqrt{2}(\ifrac{x}{5% })^{{5/4}}\right)\left(\mathop{\cos\/}\nolimits\!\left(2\sqrt{2}(\ifrac{x}{5})% ^{{5/4}}-\tfrac{1}{8}\pi\right)+\mathop{o\/}\nolimits\!\left(1\right)\right),&% x\to+\infty,\\ \mathop{\cos\/}\nolimits\!\left(4(\ifrac{|x|}{5})^{{5/4}}-\tfrac{1}{4}\pi% \right)+\mathop{o\/}\nolimits\!\left(1\right),&x\to-\infty.\end{cases}$

Symbols:: $\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than and : real parameter
Referenced by:: §36.11
Permalink:: http://dlmf.nist.gov/36.11.E4
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36.11.5 $\mathop{\Psi_{{3}}\/}\nolimits\!\left(0,y,0\right)={\mathop{\Psi_{{3}}\/}% \nolimits^{{*}}}(0,-y,0)=\mathop{\exp\/}\nolimits\!\left(\tfrac{1}{4}i\pi% \right)\sqrt{\ifrac{\pi}{y}}\left(1-(i/{\sqrt{3}})\mathop{\exp\/}\nolimits\!% \left(\tfrac{3}{2}i(\ifrac{2y}{5})^{{5/3}}\right)+\mathop{o\/}\nolimits\!\left% (1\right)\right),$ $y\to+\infty$ .

Symbols:: $\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than and : real parameter
Permalink:: http://dlmf.nist.gov/36.11.E5
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36.11.6 $\mathop{\Psi_{{3}}\/}\nolimits\!\left(0,0,z\right)=\frac{\mathop{\Gamma\/}% \nolimits\!\left(\tfrac{1}{3}\right)}{|z|^{{1/3}}\sqrt{3}}+\begin{cases}% \mathop{o\/}\nolimits\!\left(1\right),&z\to+\infty,\\ \dfrac{2\sqrt{\pi}5^{{1/4}}}{(3|z|)^{{3/4}}}\left(\mathop{\cos\/}\nolimits\!% \left(\dfrac{2}{3}\left(\dfrac{3|z|}{5}\right)^{{5/2}}-\dfrac{1}{4}\pi\right)+% \mathop{o\/}\nolimits\!\left(1\right)\right),&z\to-\infty.\end{cases}$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than and : real parameter
Permalink:: http://dlmf.nist.gov/36.11.E6
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36.11.7 $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(0,0,z\right)=\frac{\pi}{z}% \left(i+\sqrt{3}\mathop{\exp\/}\nolimits\!\left(\frac{4}{27}iz^{3}\right)+% \mathop{o\/}\nolimits\!\left(1\right)\right),$ $z\to\pm\infty$ ,

Symbols:: $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than and : real parameter
Permalink:: http://dlmf.nist.gov/36.11.E7
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36.11.8 $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(0,0,z\right)=\frac{2\pi}{z}% \left(1-\frac{i}{\sqrt{3}}\mathop{\exp\/}\nolimits\!\left(\frac{1}{27}iz^{3}% \right)+\mathop{o\/}\nolimits\!\left(1\right)\right),$ $z\to\pm\infty$ .

Symbols:: $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than and : real parameter
Permalink:: http://dlmf.nist.gov/36.11.E8
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