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1. The intuition is simple. Consider the example $f(z)=\sum_{n=0}^\infty z^{2^n}.$ This function satisfies the functional equation $f(z^2)=z^2+f(z)$. On the positive ray, we evidently have $f(r)\to\infty$ as $r\to 1$, and the functional equation shows that the same must happen on all rays $\{ re^{i\theta}:0<r<1\}$ for $\theta=k/2^n$. Since these rays are dense, all boundary points of the unit disk must be singular. Hadamard noticed that the special arithmetic nature of the sequence $2^n$ is irrelevant here it is enough to assume that $ \liminf m_{n+1}/m_n\geq 1.$ This is an implementation of the general principle: a lacunary series behaves in the same way in all directions. So if the radius of convergence is $R<\infty$ then all points $Re^{i\theta}$ must be singular.

2. The final form of this theorem is due to E. Fabry, and it is called "Fabry's gap theorem", which implies, for example that $\sum_{n=0}^\infty z^{n^2}$ is singular at every boundary point of the circle of convergence. This gap theorem is in turn is a very special case of "Fabry's General Theorem". The best source for all of this is the German book of L. Bieberbach, Analytische Fortsetzung, Springer 1955, except that it is somewhat out of date. He describes the story of Fabry's General Theorem and related results in great detail. For a modern exposition of Fabry's theorems in English, I recommend my papers

MR2595767 Eremenko, Alexandre, Densities in Fabry's theorem. Illinois J. Math. 52 (2008), no. 4, 1277–1290, and

MR2431054 Eremenko, Alexandre, A version of Fabry's theorem for power series with regularly varying coefficients. Proc. Amer. Math. Soc. 136 (2008), no. 12, 4389–4394.

Remark. Alexandr Ostrowski was 5 years old when Fabry published his general theorem. So it is unclear why the Wikipedia author calls it "Hadamard-Ostrowski".

Remark 2. A complex analyst will not describe such behavior as "badly behaved". Anyway, this behavior is typical for analytic functions, both in the sense of Baire category and in the sense of measure.

1. The intuition is simple. Consider the example $f(z)=\sum_{n=0}^\infty z^{2^n}.$ This function satisfies the functional equation $f(z^2)=z^2+f(z)$. On the positive ray, we evidently have $f(r)\to\infty$ as $r\to 1$, and the functional equation shows that the same must happen on all rays $\{ re^{i\theta}:0<r<1\}$ for $\theta=k/2^n$. Since these rays are dense, all boundary points of the unit disk must be singular. Hadamard noticed that the special arithmetic nature of the sequence $2^n$ is irrelevant here it is enough to assume that $ \liminf m_{n+1}/m_n>1$. This is an implementation of the general principle: a lacunary series behaves in the same way in all directions. So if the radius of convergence is $R<\infty$ then all points $Re^{i\theta}$ must be singular.

2. The final form of this theorem is due to E. Fabry, and it is called "Fabry's gap theorem", which implies, for example that $\sum_{n=0}^\infty z^{n^2}$ is singular at every boundary point of the circle of convergence. This gap theorem is in turn is a very special case of "Fabry's General Theorem". The best source for all of this is the German book of L. Bieberbach, Analytische Fortsetzung, Springer 1955, except that it is somewhat out of date. He describes the story of Fabry's General Theorem and related results in great detail. For a modern exposition of Fabry's theorems in English, I recommend my papers

MR2595767 Eremenko, Alexandre, Densities in Fabry's theorem. Illinois J. Math. 52 (2008), no. 4, 1277–1290, and

MR2431054 Eremenko, Alexandre, A version of Fabry's theorem for power series with regularly varying coefficients. Proc. Amer. Math. Soc. 136 (2008), no. 12, 4389–4394.

Remark. Alexandr Ostrowski was 5 years old when Fabry published his general theorem. So it is unclear why the Wikipedia author calls it "Hadamard-Ostrowski".

Remark 2. A complex analyst will not describe such behavior as "badly behaved". Anyway, this behavior is typical for analytic functions, both in the sense of Baire category and in the sense of measure.

1. The intuition is simple. Consider the example $f(z)=\sum_{n=0}^\infty z^{2^n}.$ This function satisfies the functional equation $f(z^2)=z^2+f(z)$. On the positive ray, we evidently have $f(r)\to\infty$ as $r\to 1$, and the functional equation shows that the same must happen on all rays $\{ re^{i\theta}:0<r<1\}$ for $\theta=k/2^n$. Since these rays are dense, all boundary points of the unit disk must be singular. Hadamard noticed that the special arithmetic nature of the sequence $2^n$ is irrelevant here it is enough to assume that $ \liminf m_{n+1}/m_n\geq 1.$ This is an implementation of the general principle: a lacunary series behaves in the same way in all directions. So if the radius of convergence is $R<\infty$ then all points $Re^{i\theta}$ must be singular.

2. The final form of this theorem is due to E. Fabry, and it is called "Fabry's gap theorem", which implies, for example that $\sum_{n=0}^\infty z^{n^2}$ is singular at every boundary point of the circle of convergence. This gap theorem is in turn is a very special case of "Fabry's General Theorem". The best source for all of this is the German book of L. Bieberbach, Analytische Fortsetzung, Springer 1955, except that it is somewhat out of date. He describes the story of Fabry's General Theorem and related results in great detail. For a modern exposition of Fabry's theorems in English, I recommend my papers

MR2595767 Eremenko, Alexandre, Densities in Fabry's theorem. Illinois J. Math. 52 (2008), no. 4, 1277–1290, and

MR2431054 Eremenko, Alexandre, A version of Fabry's theorem for power series with regularly varying coefficients. Proc. Amer. Math. Soc. 136 (2008), no. 12, 4389–4394.

Remark. Alexandr Ostrowski was 5 years old when Fabry published his general theorem. So it is unclear why the Wikipedia author calls it "Hadamard-Ostrowski".

1. The intuition is simple. Consider the example $f(z)=\sum_{n=0}^\infty z^{2^n}.$ This function satisfies the functional equation $f(z^2)=z^2+f(z)$. On the positive ray, we evidently have $f(r)\to\infty$ as $r\to 1$, and the functional equation shows that the same must happen on all rays $\{ re^{i\theta}:0<r<1\}$ for $\theta=k/2^n$. Since these rays are dense, all boundary points of the unit disk must be singular. Hadamard noticed that the special arithmetic nature of the sequence $2^n$ is irrelevant here it is enough to assume that $ \liminf m_{n+1}/m_n\geq 1.$ This is an implementation of the general principle: a lacunary series behaves in the same way in all directions. So if the radius of convergence is $R<\infty$ then all points $Re^{i\theta}$ must be singular.

2. The final form of this theorem is due to E. Fabry, and it is called "Fabry's gap theorem", which implies, for example that $\sum_{n=0}^\infty z^{n^2}$ is singular at every boundary point of the circle of convergence. This gap theorem is in turn is a very special case of "Fabry's General Theorem". The best source for all of this is the German book of L. Bieberbach, Analytische Fortsetzung, Springer 1955, except that it is somewhat out of date. He describes the story of Fabry's General Theorem and related results in great detail. For a modern exposition of Fabry's theorems in English, I recommend my papers

MR2595767 Eremenko, Alexandre, Densities in Fabry's theorem. Illinois J. Math. 52 (2008), no. 4, 1277–1290, and

MR2431054 Eremenko, Alexandre, A version of Fabry's theorem for power series with regularly varying coefficients. Proc. Amer. Math. Soc. 136 (2008), no. 12, 4389–4394.

Remark. Alexandr Ostrowski was 5 years old when Fabry published his general theorem. So it is unclear why the Wikipedia author calls it "Hadamard-Ostrowski".

Remark 2. A complex analyst will not describe such behavior as "badly behaved". Anyway, this behavior is typical for analytic functions, both in the sense of Baire category and in the sense of measure.

1. The intuition is simple. Consider the example $f(z)=\sum_{n=0}^\infty z^{2^n}.$ This function satisfies the functional equation $f(z^2)=1+f(z)$. On the positive ray, we evidently have $f(r)\to\infty$ as $r\to 1$, and the functional equation shows that the same must happen on all rays $\{ re^{i\theta}:0<r<1\}$ for $\theta=k/2^n$. Since these rays are dense, all boundary points of the unit disk must be singular. Hadamard noticed that the special arithmetic nature of the sequence $2^n$ is irrelevant here it is enough to assume that $ \liminf m_{n+1}/m_n\geq 1.$ This is an implementation of the general principle: a lacunary series behaves in the same way in all directions. So if the radius of convergence is $R<\infty$ then all points $Re^{i\theta}$ must be singular.

2. The final form of this theorem is due to E. Fabry, and it is called "Fabry's gap theorem", which implies, for example that $\sum_{n=0}^\infty z^{n^2}$ is singular at every boundary point of the circle of convergence. This gap theorem is in turn is a very special case of "Fabry's General Theorem". The best source for all of this is the German book of L. Bieberbach, Analytische Fortsetzung, Springer 1955, except that it is somewhat out of date. He describes the story of Fabry's General Theorem and related results in great detail. For a modern exposition of Fabry's theorems in English, I recommend my papers

MR2595767 Eremenko, Alexandre, Densities in Fabry's theorem. Illinois J. Math. 52 (2008), no. 4, 1277–1290, and

MR2431054 Eremenko, Alexandre, A version of Fabry's theorem for power series with regularly varying coefficients. Proc. Amer. Math. Soc. 136 (2008), no. 12, 4389–4394.

Remark. Alexandr Ostrowski was 5 years old when Fabry published his general theorem. So it is unclear why the Wikipedia author calls it "Hadamard-Ostrowski".

1. The intuition is simple. Consider the example $f(z)=\sum_{n=0}^\infty z^{2^n}.$ This function satisfies the functional equation $f(z^2)=z^2+f(z)$. On the positive ray, we evidently have $f(r)\to\infty$ as $r\to 1$, and the functional equation shows that the same must happen on all rays $\{ re^{i\theta}:0<r<1\}$ for $\theta=k/2^n$. Since these rays are dense, all boundary points of the unit disk must be singular. Hadamard noticed that the special arithmetic nature of the sequence $2^n$ is irrelevant here it is enough to assume that $ \liminf m_{n+1}/m_n\geq 1.$ This is an implementation of the general principle: a lacunary series behaves in the same way in all directions. So if the radius of convergence is $R<\infty$ then all points $Re^{i\theta}$ must be singular.

2. The final form of this theorem is due to E. Fabry, and it is called "Fabry's gap theorem", which implies, for example that $\sum_{n=0}^\infty z^{n^2}$ is singular at every boundary point of the circle of convergence. This gap theorem is in turn is a very special case of "Fabry's General Theorem". The best source for all of this is the German book of L. Bieberbach, Analytische Fortsetzung, Springer 1955, except that it is somewhat out of date. He describes the story of Fabry's General Theorem and related results in great detail. For a modern exposition of Fabry's theorems in English, I recommend my papers

MR2595767 Eremenko, Alexandre, Densities in Fabry's theorem. Illinois J. Math. 52 (2008), no. 4, 1277–1290, and

MR2431054 Eremenko, Alexandre, A version of Fabry's theorem for power series with regularly varying coefficients. Proc. Amer. Math. Soc. 136 (2008), no. 12, 4389–4394.

Remark. Alexandr Ostrowski was 5 years old when Fabry published his general theorem. So it is unclear why the Wikipedia author calls it "Hadamard-Ostrowski".