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Specific values of Bessel functions

Table of contents: Trigonometric cases - Hyperbolic cases - Airy function cases

Related topics: Bessel functions

Trigonometric cases

621a9b
J1/2 ⁣(z)=(2zπ)1/2cos ⁣(z)zJ_{-1 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\cos\!\left(z\right)}{z}
121b21
J1/2 ⁣(z)=(2zπ)1/2sin ⁣(z)zJ_{1 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\sin\!\left(z\right)}{z}
a2a294
J3/2 ⁣(z)=(2zπ)1/2(sin ⁣(z)z2cos ⁣(z)z)J_{3 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \left(\frac{\sin\!\left(z\right)}{{z}^{2}} - \frac{\cos\!\left(z\right)}{z}\right)
5679f2
Y1/2 ⁣(z)=(2zπ)1/2sin ⁣(z)zY_{-1 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\sin\!\left(z\right)}{z}
4dfd41
Y1/2 ⁣(z)=(2zπ)1/2cos ⁣(z)zY_{1 / 2}\!\left(z\right) = -{\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\cos\!\left(z\right)}{z}
8472cc
Y3/2 ⁣(z)=(2zπ)1/2(cos ⁣(z)z2+sin ⁣(z)z)Y_{3 / 2}\!\left(z\right) = -{\left(\frac{2 z}{\pi}\right)}^{1 / 2} \left(\frac{\cos\!\left(z\right)}{{z}^{2}} + \frac{\sin\!\left(z\right)}{z}\right)

Hyperbolic cases

5d9c43
I1/2 ⁣(z)=(2zπ)1/2cosh ⁣(z)zI_{-1 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\cosh\!\left(z\right)}{z}
a59981
I1/2 ⁣(z)=(2zπ)1/2sinh ⁣(z)zI_{1 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\sinh\!\left(z\right)}{z}
65647f
I3/2 ⁣(z)=(2zπ)1/2(cosh ⁣(z)zsinh ⁣(z)z2)I_{3 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \left(\frac{\cosh\!\left(z\right)}{z} - \frac{\sinh\!\left(z\right)}{{z}^{2}}\right)
7ac286
K1/2 ⁣(z)=(πz2)1/2ezzK_{-1 / 2}\!\left(z\right) = {\left(\frac{\pi z}{2}\right)}^{1 / 2} \frac{{e}^{-z}}{z}
d1f5c5
K1/2 ⁣(z)=(πz2)1/2ezzK_{1 / 2}\!\left(z\right) = {\left(\frac{\pi z}{2}\right)}^{1 / 2} \frac{{e}^{-z}}{z}
0c09cc
K3/2 ⁣(z)=(πz2)1/2ez(1z+1z2)K_{3 / 2}\!\left(z\right) = {\left(\frac{\pi z}{2}\right)}^{1 / 2} {e}^{-z} \left(\frac{1}{z} + \frac{1}{{z}^{2}}\right)

Airy function cases

Related topics: Airy functions

685892
J1/3 ⁣(z)=12ω(3Ai ⁣(ω2)+3Bi ⁣(ω2))   where ω=(3z2)1/3J_{-1 / 3}\!\left(z\right) = \frac{1}{2 \omega} \left(3 \operatorname{Ai}\!\left(-{\omega}^{2}\right) + \sqrt{3} \operatorname{Bi}\!\left(-{\omega}^{2}\right)\right)\; \text{ where } \omega = {\left(\frac{3 z}{2}\right)}^{1 / 3}
d39c46
J1/3 ⁣(z)=12ω(3Ai ⁣(ω2)3Bi ⁣(ω2))   where ω=(3z2)1/3J_{1 / 3}\!\left(z\right) = \frac{1}{2 \omega} \left(3 \operatorname{Ai}\!\left(-{\omega}^{2}\right) - \sqrt{3} \operatorname{Bi}\!\left(-{\omega}^{2}\right)\right)\; \text{ where } \omega = {\left(\frac{3 z}{2}\right)}^{1 / 3}
e72e96
J2/3 ⁣(z)=12ω2(3Ai ⁣(ω2)+3Bi ⁣(w2))   where ω=(3z2)1/3J_{2 / 3}\!\left(z\right) = \frac{1}{2 {\omega}^{2}} \left(3 \operatorname{Ai}'\!\left(-{\omega}^{2}\right) + \sqrt{3} \operatorname{Bi}'\!\left(-{w}^{2}\right)\right)\; \text{ where } \omega = {\left(\frac{3 z}{2}\right)}^{1 / 3}
fda595
K1/3 ⁣(z)=3πωAi ⁣(ω2)   where ω=(3z2)1/3K_{-1 / 3}\!\left(z\right) = \frac{\sqrt{3} \pi}{\omega} \operatorname{Ai}\!\left({\omega}^{2}\right)\; \text{ where } \omega = {\left(\frac{3 z}{2}\right)}^{1 / 3}
49d754
K1/3 ⁣(z)=3πωAi ⁣(ω2)   where ω=(3z2)1/3K_{1 / 3}\!\left(z\right) = \frac{\sqrt{3} \pi}{\omega} \operatorname{Ai}\!\left({\omega}^{2}\right)\; \text{ where } \omega = {\left(\frac{3 z}{2}\right)}^{1 / 3}
c362e8
K2/3 ⁣(z)=3πω2Ai ⁣(ω2)   where ω=(3z2)1/3K_{2 / 3}\!\left(z\right) = -\frac{\sqrt{3} \pi}{{\omega}^{2}} \operatorname{Ai}'\!\left({\omega}^{2}\right)\; \text{ where } \omega = {\left(\frac{3 z}{2}\right)}^{1 / 3}

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2019-08-25 15:30:03.056001 UTC