Gauss hypergeometric function derivative

[fred_91]

Homework Statement

I want to differentiate the Gauss hypergeometric function:
$_2F_1[a,b;c;\frac{k-x}{z-x}]$
with respect to z

Homework Equations

The derivative of
$_2F_1[a,b;c;z]$
with respect to z is:
$\frac{ab}{c} _2F_1[1+a,1+b;1+c;z]$

The Attempt at a Solution

Can I treat this as any other function, i.e., the same as with z in the fourth parameter but multiplied by the derivative of the fourth parameter:

$\frac{ab}{c} _2F_1[a,b;c;\frac{k-x}{z-x}]\frac{-(k-x)}{(z-x)^2}$

If the fourth parameter was sin(z). Would the derivative of
$_2F_1[a,b;c;\sin(z)]$
with respect to z
be
$\frac{ab}{c} _2F_1[1+a,1+b;1+c;sin(z)]cos(z)$?

Thank you in advance.

 

[mighty2000]

I would like to offer some guidance on how to approach differentiating the Gauss hypergeometric function.

Firstly, it is important to understand the properties and rules of differentiation. In this case, we can use the chain rule to differentiate the function with respect to z. This means that we need to consider the derivative of the fourth parameter (in this case, \frac{k-x}{z-x}) and the derivative of the function itself.

The derivative of the fourth parameter can be calculated using the quotient rule, which gives us:

\frac{d}{dz}(\frac{k-x}{z-x}) = \frac{(z-x)\frac{d}{dz}(k-x) - (k-x)\frac{d}{dz}(z-x)}{(z-x)^2} = \frac{k-x}{(z-x)^2}

Now, using the chain rule, we can write the derivative of the function as:

\frac{d}{dz}(_2F_1[a,b;c;\frac{k-x}{z-x}]) = \frac{d}{dz}(_2F_1[a,b;c;u]) \frac{d}{dz}(\frac{k-x}{z-x}) = \frac{ab}{c} _2F_1[1+a,1+b;1+c;\frac{k-x}{z-x}] \frac{k-x}{(z-x)^2}

Therefore, the final answer would be:

\frac{ab}{c} _2F_1[1+a,1+b;1+c;\frac{k-x}{z-x}] \frac{k-x}{(z-x)^2}

To answer your second question, the derivative of the function with respect to z when the fourth parameter is sin(z) would be:

\frac{ab}{c} _2F_1[1+a,1+b;1+c;\sin(z)]\cos(z)

I hope this helps you in your understanding of differentiating the Gauss hypergeometric function. Remember to always carefully consider the properties and rules of differentiation when approaching a new function.