Alternating harmonic series of modified Bessel functions
In Abramowitz and Stegun §9.6.33ff. there are a bunch of identities listed involving the modified Bessel function. Relevant for my fooling question are probably these ones:
$$begin{align}
1 &= I_0(z) - 2I_2(z) + 2I_4(z) - 2I_6(z) + dots \
\
e^z &= I_0(z) + 2I_1(z) + 2I_2(z) + 2I_3(z) + dots \
e^{-z} &= I_0(z) - 2I_1(z) + 2I_2(z) - 2I_3(z) + dots \
\
cosh{z} &= phantom{2}I_0(z) + 2I_2(z) + 2I_4(z) + 2I_6(z) + dots \
sinh{z} &= 2I_1(z) + 2I_3(z) + 2I_5(z) + 2I_7(z) + dots \
end{align}$$
Now, I came across the following series
$$f(z) =I_1(z) - frac{I_3(z)}{3} + frac{I_5(z)}{5} - frac{I_7(z)}{7} + dots$$
and was wondering, whether one can deduce a nice analytical expression for $f(z)$ (not necessarily using the identities above).
sequences-and-series bessel-functions
asked 2 days ago
chickenNinja123
959
add a comment |
In Abramowitz and Stegun §9.6.33ff. there are a bunch of identities listed involving the modified Bessel function. Relevant for my fooling question are probably these ones:
$$begin{align}
1 &= I_0(z) - 2I_2(z) + 2I_4(z) - 2I_6(z) + dots \
\
e^z &= I_0(z) + 2I_1(z) + 2I_2(z) + 2I_3(z) + dots \
e^{-z} &= I_0(z) - 2I_1(z) + 2I_2(z) - 2I_3(z) + dots \
\
cosh{z} &= phantom{2}I_0(z) + 2I_2(z) + 2I_4(z) + 2I_6(z) + dots \
sinh{z} &= 2I_1(z) + 2I_3(z) + 2I_5(z) + 2I_7(z) + dots \
end{align}$$
Now, I came across the following series
$$f(z) =I_1(z) - frac{I_3(z)}{3} + frac{I_5(z)}{5} - frac{I_7(z)}{7} + dots$$
and was wondering, whether one can deduce a nice analytical expression for $f(z)$ (not necessarily using the identities above).
sequences-and-series bessel-functions
asked 2 days ago
chickenNinja123
959
add a comment |
In Abramowitz and Stegun §9.6.33ff. there are a bunch of identities listed involving the modified Bessel function. Relevant for my fooling question are probably these ones:
$$begin{align}
1 &= I_0(z) - 2I_2(z) + 2I_4(z) - 2I_6(z) + dots \
\
e^z &= I_0(z) + 2I_1(z) + 2I_2(z) + 2I_3(z) + dots \
e^{-z} &= I_0(z) - 2I_1(z) + 2I_2(z) - 2I_3(z) + dots \
\
cosh{z} &= phantom{2}I_0(z) + 2I_2(z) + 2I_4(z) + 2I_6(z) + dots \
sinh{z} &= 2I_1(z) + 2I_3(z) + 2I_5(z) + 2I_7(z) + dots \
end{align}$$
Now, I came across the following series
$$f(z) =I_1(z) - frac{I_3(z)}{3} + frac{I_5(z)}{5} - frac{I_7(z)}{7} + dots$$
and was wondering, whether one can deduce a nice analytical expression for $f(z)$ (not necessarily using the identities above).
sequences-and-series bessel-functions
asked 2 days ago
chickenNinja123
959
In Abramowitz and Stegun §9.6.33ff. there are a bunch of identities listed involving the modified Bessel function. Relevant for my fooling question are probably these ones:
$$begin{align}
1 &= I_0(z) - 2I_2(z) + 2I_4(z) - 2I_6(z) + dots \
\
e^z &= I_0(z) + 2I_1(z) + 2I_2(z) + 2I_3(z) + dots \
e^{-z} &= I_0(z) - 2I_1(z) + 2I_2(z) - 2I_3(z) + dots \
\
cosh{z} &= phantom{2}I_0(z) + 2I_2(z) + 2I_4(z) + 2I_6(z) + dots \
sinh{z} &= 2I_1(z) + 2I_3(z) + 2I_5(z) + 2I_7(z) + dots \
end{align}$$
Now, I came across the following series
$$f(z) =I_1(z) - frac{I_3(z)}{3} + frac{I_5(z)}{5} - frac{I_7(z)}{7} + dots$$
and was wondering, whether one can deduce a nice analytical expression for $f(z)$ (not necessarily using the identities above).
sequences-and-series bessel-functions
sequences-and-series bessel-functions
asked 2 days ago
chickenNinja123
959
asked 2 days ago
chickenNinja123
959
asked 2 days ago
chickenNinja123
959
asked 2 days ago
chickenNinja123
959
asked 2 days ago
chickenNinja123
959
959
add a comment |
add a comment |
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