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

Sinc function