Define $F(z)=lim_{nrightarrowinfty}frac{n!n^z}{z(1+z)…(n+z)}$
Define $$F(z)=lim_{nrightarrowinfty}frac{n!n^z}{z(1+z)...(n+z)}$$
Show that $F$ defines a meromorphic function on $mathbb{C}$ and identify its poles. Also show that $F(n)=(n-1)!$
I'm not sure how to show that $F$ is a meromorphic function. However, we have
$$begin{align*}
F(n)&=lim_{nrightarrowinfty}frac{n!n^n}{n(1+n)...(2n)}\
&=lim_{nrightarrowinfty}frac{(n-1)!n!n^n}{(2n)!}
end{align*}$$
And I'm also stuck here. Any input is appreciated.
complex-analysis meromorphic-functions
asked 2 days ago
Ya G
47629
add a comment |
Define $$F(z)=lim_{nrightarrowinfty}frac{n!n^z}{z(1+z)...(n+z)}$$
Show that $F$ defines a meromorphic function on $mathbb{C}$ and identify its poles. Also show that $F(n)=(n-1)!$
I'm not sure how to show that $F$ is a meromorphic function. However, we have
$$begin{align*}
F(n)&=lim_{nrightarrowinfty}frac{n!n^n}{n(1+n)...(2n)}\
&=lim_{nrightarrowinfty}frac{(n-1)!n!n^n}{(2n)!}
end{align*}$$
And I'm also stuck here. Any input is appreciated.
complex-analysis meromorphic-functions
asked 2 days ago
Ya G
47629
2
the two $n$ are different in each expression
– Jakobian
2 days ago
@JakobianWhat do you mean?
– Ya G
2 days ago
2
$F(n) = lim_{ktoinfty} frac{k!k^n}{n(n+1)...(n+k)}$
– Jakobian
2 days ago
$F$ is actually the gamma function
– Nick
2 days ago
To get the value of $F(n),$ Stirling's Approximation will suffice. To show it is meromorphic, you have a number of choices. Clearly the poles are at $N_{le 0},$ so you could simply show uniform convergence on compact subsets excluding those points and use a bit of theory. Alternatively, see this answer. Alternatively, you could express the limit in terms of the Hadamard Product and conclude. It also depends on a bit on what definition of Meromorphic you are using.
– Brevan Ellefsen
2 days ago
add a comment |
Define $$F(z)=lim_{nrightarrowinfty}frac{n!n^z}{z(1+z)...(n+z)}$$
Show that $F$ defines a meromorphic function on $mathbb{C}$ and identify its poles. Also show that $F(n)=(n-1)!$
I'm not sure how to show that $F$ is a meromorphic function. However, we have
$$begin{align*}
F(n)&=lim_{nrightarrowinfty}frac{n!n^n}{n(1+n)...(2n)}\
&=lim_{nrightarrowinfty}frac{(n-1)!n!n^n}{(2n)!}
end{align*}$$
And I'm also stuck here. Any input is appreciated.
complex-analysis meromorphic-functions
asked 2 days ago
Ya G
47629
Define $$F(z)=lim_{nrightarrowinfty}frac{n!n^z}{z(1+z)...(n+z)}$$
Show that $F$ defines a meromorphic function on $mathbb{C}$ and identify its poles. Also show that $F(n)=(n-1)!$
I'm not sure how to show that $F$ is a meromorphic function. However, we have
$$begin{align*}
F(n)&=lim_{nrightarrowinfty}frac{n!n^n}{n(1+n)...(2n)}\
&=lim_{nrightarrowinfty}frac{(n-1)!n!n^n}{(2n)!}
end{align*}$$
And I'm also stuck here. Any input is appreciated.
complex-analysis meromorphic-functions
complex-analysis meromorphic-functions
asked 2 days ago
Ya G
47629
asked 2 days ago
Ya G
47629
asked 2 days ago
Ya G
47629
asked 2 days ago
Ya G
47629
asked 2 days ago
Ya G
47629
47629
2
the two $n$ are different in each expression
– Jakobian
2 days ago
@JakobianWhat do you mean?
– Ya G
2 days ago
2
$F(n) = lim_{ktoinfty} frac{k!k^n}{n(n+1)...(n+k)}$
– Jakobian
2 days ago
$F$ is actually the gamma function
– Nick
2 days ago
To get the value of $F(n),$ Stirling's Approximation will suffice. To show it is meromorphic, you have a number of choices. Clearly the poles are at $N_{le 0},$ so you could simply show uniform convergence on compact subsets excluding those points and use a bit of theory. Alternatively, see this answer. Alternatively, you could express the limit in terms of the Hadamard Product and conclude. It also depends on a bit on what definition of Meromorphic you are using.
– Brevan Ellefsen
2 days ago
add a comment |
2
the two $n$ are different in each expression
– Jakobian
2 days ago
@JakobianWhat do you mean?
– Ya G
2 days ago
2
$F(n) = lim_{ktoinfty} frac{k!k^n}{n(n+1)...(n+k)}$
– Jakobian
2 days ago
$F$ is actually the gamma function
– Nick
2 days ago
To get the value of $F(n),$ Stirling's Approximation will suffice. To show it is meromorphic, you have a number of choices. Clearly the poles are at $N_{le 0},$ so you could simply show uniform convergence on compact subsets excluding those points and use a bit of theory. Alternatively, see this answer. Alternatively, you could express the limit in terms of the Hadamard Product and conclude. It also depends on a bit on what definition of Meromorphic you are using.
– Brevan Ellefsen
2 days ago
2
2
the two $n$ are different in each expression
– Jakobian
2 days ago
the two $n$ are different in each expression
– Jakobian
2 days ago
@JakobianWhat do you mean?
– Ya G
2 days ago
@JakobianWhat do you mean?
– Ya G
2 days ago
2
2
$F(n) = lim_{ktoinfty} frac{k!k^n}{n(n+1)...(n+k)}$
– Jakobian
2 days ago
$F(n) = lim_{ktoinfty} frac{k!k^n}{n(n+1)...(n+k)}$
– Jakobian
2 days ago
$F$ is actually the gamma function
– Nick
2 days ago
$F$ is actually the gamma function
– Nick
2 days ago
To get the value of $F(n),$ Stirling's Approximation will suffice. To show it is meromorphic, you have a number of choices. Clearly the poles are at $N_{le 0},$ so you could simply show uniform convergence on compact subsets excluding those points and use a bit of theory. Alternatively, see this answer. Alternatively, you could express the limit in terms of the Hadamard Product and conclude. It also depends on a bit on what definition of Meromorphic you are using.
– Brevan Ellefsen
2 days ago
To get the value of $F(n),$ Stirling's Approximation will suffice. To show it is meromorphic, you have a number of choices. Clearly the poles are at $N_{le 0},$ so you could simply show uniform convergence on compact subsets excluding those points and use a bit of theory. Alternatively, see this answer. Alternatively, you could express the limit in terms of the Hadamard Product and conclude. It also depends on a bit on what definition of Meromorphic you are using.
– Brevan Ellefsen
2 days ago
add a comment |
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2
the two $n$ are different in each expression
– Jakobian
2 days ago
@JakobianWhat do you mean?
– Ya G
2 days ago
2
$F(n) = lim_{ktoinfty} frac{k!k^n}{n(n+1)...(n+k)}$
– Jakobian
2 days ago
$F$ is actually the gamma function
– Nick
2 days ago
To get the value of $F(n),$ Stirling's Approximation will suffice. To show it is meromorphic, you have a number of choices. Clearly the poles are at $N_{le 0},$ so you could simply show uniform convergence on compact subsets excluding those points and use a bit of theory. Alternatively, see this answer. Alternatively, you could express the limit in terms of the Hadamard Product and conclude. It also depends on a bit on what definition of Meromorphic you are using.
– Brevan Ellefsen
2 days ago