The analyticity of holomorphic functions

In a comprehensive analysis, the function $f$ ${\ displaystyle f}$ $f$ complex variable called

holomorphic at the point $a$ ${\ displaystyle a}$ $a$ if it is differentiable in some open neighborhood
analytical at the point ${\textstyle a}$ ${\ textstyle a}$ ${\ textstyle a}$ if there exists some neighborhood of the point $a$ ${\ displaystyle a}$ $a$ , wherein $f$ ${\ displaystyle f}$ $f$ coincides with a convergent power series

f(z)=\sum _{n=0}^{\infty }c_{n}(z-a)^{n}

{\ displaystyle f (z) = \ sum _ {n = 0} ^ {\ infty} c_ {n} (za) ^ {n}}

{\ displaystyle f (z) = \ sum _ {n = 0} ^ {\ infty} c_ {n} (z-a) ^ {n}}

One of the most important results of complex analysis is the theorem that holomorphic functions are analytic . Corollaries to this theorem include, among others, the following results:

uniqueness theorem: two holomorphic functions whose values coincide at each point of the set $S$ ${\ displaystyle S}$ $S$ (which has a limit point inside the intersection of the domains of definition of functions) also coincide in any open connected subset of their domains of definition that contains $S$ ${\ displaystyle S}$ $S$ .
since the power series is infinitely differentiable, the corresponding holomorphic function is also infinitely differentiable (in contrast to the case of a differentiable real function).
the radius of convergence always coincides with the distance from the center a to the nearest singular point . If there are none, (i.e. if $f$ ${\ displaystyle f}$ $f$ Is an entire function ), then the radius of convergence is equal to infinity.
an entire holomorphic function cannot be compactly supported, i.e. cannot have as a (compact) support a connected open subset of the complex plane.

Proof

The proof, first proposed by Cauchy, is based on the Cauchy integral formula and on the expansion in the power series of the expression

{\dfrac {1}{w-z}}

{\ displaystyle {\ dfrac {1} {wz}}}

{\dfrac {1}{w-z}}

Let be $D$ ${\ displaystyle D}$ $D$ denotes an open disk centered at $a$ ${\ displaystyle a}$ $a$ . Let's pretend that $f$ ${\ displaystyle f}$ $f$ is differentiable everywhere in the open neighborhood of the closure $D$ ${\ displaystyle D}$ $D$ . Let be $C$ ${\ displaystyle C}$ $C$ denotes a positively oriented circle which is the boundary $D$ ${\ displaystyle D}$ $D$ , a $z$ ${\ displaystyle z}$ $z$ - point in $D$ ${\ displaystyle D}$ $D$ . Starting with the Cauchy integral formula, we can write

{\begin{aligned}f(z)&{}={1 \over 2\pi i}\int _{C}{f(w) \over w-z}\,\mathrm {d} w\\[10pt]&{}={1 \over 2\pi i}\int _{C}{f(w) \over (w-a)-(z-a)}\,\mathrm {d} w\\[10pt]&{}={1 \over 2\pi i}\int _{C}{1 \over w-a}\cdot {1 \over 1-{z-a \over w-a}}f(w)\,\mathrm {d} w\\[10pt]&{}={1 \over 2\pi i}\int _{C}{1 \over w-a}\cdot {\sum _{n=0}^{\infty }\left({z-a \over w-a}\right)^{n}}f(w)\,\mathrm {d} w\\[10pt]&{}=\sum _{n=0}^{\infty }{1 \over 2\pi i}\int _{C}{(z-a)^{n} \over (w-a)^{n+1}}f(w)\,\mathrm {d} w.\end{aligned}}

{\ displaystyle {\ begin {aligned} f (z) & {} = {1 \ over 2 \ pi i} \ int _ {C} {f (w) \ over wz} \, \ mathrm {d} w \ \ [10pt] & {} = {1 \ over 2 \ pi i} \ int _ {C} {f (w) \ over (wa) - (za)} \, \ mathrm {d} w \\ [10pt ] & {} = {1 \ over 2 \ pi i} \ int _ {C} {1 \ over wa} \ cdot {1 \ over 1- {za \ over wa}} f (w) \, \ mathrm { d} w \\ [10pt] & {} = {1 \ over 2 \ pi i} \ int _ {C} {1 \ over wa} \ cdot {\ sum _ {n = 0} ^ {\ infty} \ left ({za \ over wa} \ right) ^ {n}} f (w) \, \ mathrm {d} w \\ [10pt] & {} = \ sum _ {n = 0} ^ {\ infty} {1 \ over 2 \ pi i} \ int _ {C} {(za) ^ {n} \ over (wa) ^ {n + 1}} f (w) \, \ mathrm {d} w. \ End {aligned}}}

{\begin{aligned}f(z)&{}={1 \over 2\pi i}\int _{C}{f(w) \over w-z}\,\mathrm {d} w\\[10pt]&{}={1 \over 2\pi i}\int _{C}{f(w) \over (w-a)-(z-a)}\,\mathrm {d} w\\[10pt]&{}={1 \over 2\pi i}\int _{C}{1 \over w-a}\cdot {1 \over 1-{z-a \over w-a}}f(w)\,\mathrm {d} w\\[10pt]&{}={1 \over 2\pi i}\int _{C}{1 \over w-a}\cdot {\sum _{n=0}^{\infty }\left({z-a \over w-a}\right)^{n}}f(w)\,\mathrm {d} w\\[10pt]&{}=\sum _{n=0}^{\infty }{1 \over 2\pi i}\int _{C}{(z-a)^{n} \over (w-a)^{n+1}}f(w)\,\mathrm {d} w.\end{aligned}}

The permutation of the integration operations and the infinite sum is valid, since the expression is limited by some positive constant $M$ ${\ displaystyle M}$ for any $w$ ${\ displaystyle w}$ on $C$ ${\ displaystyle C}$ while inequality

$\left|{\frac {z-a}{w-a}}\right|\leq r<1$ ${\ displaystyle \ left | {\ frac {za} {wa}} \ right | \ leq r <1}$ also true on $C$ ${\ displaystyle C}$ with some positive $r$ ${\ displaystyle r}$ . In this way,

\left|{(z-a)^{n} \over (w-a)^{n+1}}f(w)\right|\leq Mr^{n}

{\ displaystyle \ left | {(za) ^ {n} \ over (wa) ^ {n + 1}} f (w) \ right | \ leq Mr ^ {n}}

on $C$ ${\ displaystyle C}$ . The row converges uniformly on $C$ ${\ displaystyle C}$ on the basis of Weierstrass convergence , which means that the signs of the sum and integral can be rearranged.

Since the expression $(z-a)^{n}$ ${\ displaystyle (za) ^ {n}}$ independent of the variable, it can be taken out of the sign of the integral

f(z)=\sum _{n=0}^{\infty }(z-a)^{n}{1 \over 2\pi i}\int _{C}{f(w) \over (w-a)^{n+1}}\,\mathrm {d} w

{\ displaystyle f (z) = \ sum _ {n = 0} ^ {\ infty} (za) ^ {n} {1 \ over 2 \ pi i} \ int _ {C} {f (w) \ over (wa) ^ {n + 1}} \, \ mathrm {d} w}

Thus, the decomposition of the function $f$ ${\ displaystyle f}$ acquires the desired power-law form from $z$ ${\ displaystyle z}$ :

f(z)=\sum _{n=0}^{\infty }c_{n}(z-a)^{n}

{\ displaystyle f (z) = \ sum _ {n = 0} ^ {\ infty} c_ {n} (za) ^ {n}}

with coefficients

c_{n}={1 \over 2\pi i}\int _{C}{f(w) \over (w-a)^{n+1}}\,\mathrm {d} w.

{\ displaystyle c_ {n} = {1 \ over 2 \ pi i} \ int _ {C} {f (w) \ over (wa) ^ {n + 1}} \, \ mathrm {d} w.}

Notes

Since power series can be differentiated term by term, the above reasoning can be applied in the opposite direction to the expansion of the expression

{\frac {1}{(w-z)^{n+1}}}

{\ displaystyle {\ frac {1} {(wz) ^ {n + 1}}}}

having thus

$f^{(n)}(a)={n! \over 2\pi i}\int _{C}{f(w) \over (w-a)^{n+1}}\,dw.$ ${\ displaystyle f ^ {(n)} (a) = {n!$ $\ over 2 \ pi i} \ int _ {C} {f (w) \ over (wa) ^ {n + 1}} \, dw.}$

This is the Cauchy integral formula for derivatives. Thus, the above power series is a Taylor series of function

f

{\ displaystyle f}

.

The proof is valid only if the point $z$ ${\ displaystyle z}$ is closer to the center $a$ ${\ displaystyle a}$ drive $D$ ${\ displaystyle D}$ than to any singular point $f$ ${\ displaystyle f}$ . Consequently, the radius of convergence of the Taylor series cannot be less than the distance from $a$ ${\ displaystyle a}$ to the nearest singular point of the function. (Obviously, the radius also cannot be greater than this distance, since the power series has no singular points within their radius of convergence.)
A special case of the same theorem From the previous remark follows a special case of the uniqueness theorem for a holomorphic function. If two holomorphic functions coincide on a (possibly very small) open neighborhood $U$ ${\ displaystyle U}$ points $a$ ${\ displaystyle a}$ , then they match on the open drive $B_{d}(a)$ ${\ displaystyle B_ {d} (a)}$ where $d$ ${\ displaystyle d}$ - distance from $a$ ${\ displaystyle a}$ to the nearest singular point.

The analyticity of holomorphic functions

Proof

Notes

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