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EngWiPy
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Hello,
What is [tex]\frac{d}{dx}K_v\left(f(x)\right)=?[/tex]
Thanks in advance
What is [tex]\frac{d}{dx}K_v\left(f(x)\right)=?[/tex]
Thanks in advance
Mute said:For [itex]\nu[/itex] not necessarily an integer, [itex]C_{\nu}(y) = e^{\nu \pi i}K_{\nu}(y)[/itex] satisfies the identity
[tex]2\frac{dC_\nu}{dy} = C_{\nu-1}(y) + C_{\nu + 1}(y)[/tex]
Then let [itex]y = f(x)[/itex] and use the chain rule.
Identities like this can usually be found on wikipedia, and for something as studied and used as frequently as the Bessel Functions, are generally correct.
http://en.wikipedia.org/wiki/Bessel_function
For a more 'official' reference, see something like http://www.math.sfu.ca/~cbm/aands/page_437.htm (scans of a reference book).
S_David said:Hello,
What is [tex]\frac{d}{dx}K_v\left(f(x)\right)=?[/tex]
Thanks in advance
skyspeed said:Do you get the answer?
S_David said:I know that
[tex]z\frac{d}{dz}K_v(z)+vK_v(z)=-zK_{v-1}(z)[/tex]
but I was confused when we have more complicated arguments such as
[tex]z=\sqrt{x^2+x}[/tex].
But after your posting, I have now a simple method to move from simple to more complicated arguments. So, I can say the following:
[tex]\frac{f(z)}{f'(z)}\frac{d}{dz}K_v(f(z))+vK_v(f(z))=-f(z)K_{v-1}(f(z))[/tex]
Am I right?
Best regards
S_David said:let [tex]y=f(x)[/tex] and then use the chain rule.
The Bessel function of the second kind, denoted as Yv(x), is a special mathematical function that is used to solve differential equations. It is closely related to the Bessel function of the first kind, denoted as Jv(x), and is defined as the solution to the Bessel differential equation of the second kind.
The derivative of the Bessel function of the second kind, denoted as Y'v(x), can be expressed in terms of the Bessel function of the first kind and the Bessel function of the second kind. This derivative can be used to solve differential equations involving Bessel functions.
The Bessel function of the second kind has many applications in physics, particularly in problems involving wave propagation and diffraction. It is also used in the analysis of heat conduction and the behavior of electromagnetic fields.
The Bessel function of the second kind can be calculated using various methods, such as power series, continued fractions, and recurrence relations. It can also be computed numerically using software programs or calculators.
The Bessel function of the second kind has several important properties, including its asymptotic behavior, zeros, and integral representations. It also has a connection to other special functions, such as the modified Bessel function and the Hankel function.