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Fungrim entry: aa15f0

$z f''(z) + 2 f'(z) + {A}^{2} z f(z) = 0\; \text{ where } f(z) = {C}_{1} \operatorname{sinc}\!\left(A z\right) + {C}_{2} \frac{\cos\!\left(A z\right)}{z}$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \ne 0 \;\mathbin{\operatorname{and}}\; A \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {C}_{1} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {C}_{2} \in \mathbb{C}$
TeX:
z f''(z) + 2 f'(z) + {A}^{2} z f(z) = 0\; \text{ where } f(z) = {C}_{1} \operatorname{sinc}\!\left(A z\right) + {C}_{2} \frac{\cos\!\left(A z\right)}{z}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \ne 0 \;\mathbin{\operatorname{and}}\; A \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {C}_{1} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {C}_{2} \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
Pow${a}^{b}$ Power
Sinc$\operatorname{sinc}(z)$ Sinc function
Cos$\cos(z)$ Cosine
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("aa15f0"),
Formula(Where(Equal(Add(Add(Mul(z, ComplexDerivative(f(z), For(z, z, 2))), Mul(2, ComplexDerivative(f(z), For(z, z)))), Mul(Mul(Pow(A, 2), z), f(z))), 0), Equal(f(z), Add(Mul(Subscript(C, 1), Sinc(Mul(A, z))), Mul(Subscript(C, 2), Div(Cos(Mul(A, z)), z)))))),
Variables(C, A, z),
Assumptions(And(Element(z, CC), NotEqual(z, 0), Element(A, CC), Element(Subscript(C, 1), CC), Element(Subscript(C, 2), CC))))

Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC