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Such an integral over a simple (non-closed) curve is called the Cauchy type integral. It is convenient to define $$F(z)=\frac{1}{2\pi i}\int_\gamma \frac{f(\zeta)d\zeta}{\zeta-z},\quad z\not\in\gamma.\tag{1}$$

The curve $\gamma$ is oriented, so for our function $F$ defined in $C\backslash\gamma$ and $z\in\gamma$ different from an endpoint, we can talk of the right limit $F^+(z)$ and left limit $F^-(z)$. Then we have Sokhotski formula $$F^-(z)-F^+(z)=f(z),\quad z\in\Gamma$$ As in addition $F(z)\to0,\; z\to\infty$, this shows that your map is injective. The image consists of those analytic functions in $C\backslash\gamma$ which tend to $0$ as $z\to\infty$, have limits on both sides of $\gamma$ and the jump between these limits is an entire function (analytic in $C$). (At the end-points, the limits do not exist, and we have logarithmic singularities.)

With some regularity of $\gamma$ and $f$ (non necessary analytic) we have $$F^+(z)=F^*(z)-\frac{1}{2}f(z),\quad F^-(z)=F^*(z)+\frac{1}{2}f(z),$$ where $F^*(z)$ is defined for $z\in\gamma$ using the principal value of the integral in (1). The above formula is obtained by subtracting these two relations. But when $f$ is entire, the curve can be slightly deformed (preserving its ends), and you get rid of the regularity condition by approximating it with a smooth (or piecewise-linear) curve with the same ends.

If the curve happens to be closed then $F^+=0$ and we recover the result for the closed curve which you cite.

It is interesting that the curve $\gamma$ plays little role in the description of the image: functions from the image have analytic continuation along any curve starting and ending at $\infty$ and not passing through the endpoints of $\gamma$.

References. This material is standard and is included to every Russian undergraduate Complex Variables text, but for some strange reasons none of them is translated into English. A good reference in English which comes to my mind is a translation from the Japanese:

MR1026013
Kaneko, A. Introduction to hyperfunctions. Kluwer, Dordrecht, Tokyo, 1988.

Such an integral over a simple (non-closed) curve is called the Cauchy type integral. It is convenient to define $$F(z)=\frac{1}{2\pi i}\int_\gamma \frac{f(\zeta)d\zeta}{\zeta-z},\quad z\not\in\gamma.\tag{1}$$

The curve $\gamma$ is oriented, so for our function $F$ defined in $C\backslash\gamma$ and $z\in\gamma$ different from an endpoint, we can talk of the right limit $F^+(z)$ and left limit $F^-(z)$. Then we have Sokhotski formula $$F^-(z)-F^+(z)=f(z),\quad z\in\gamma$$ As in addition $F(z)\to0,\; z\to\infty$, this shows that your map is injective. (If we have two functions with the same jump, their difference is entire, and since it tends to zero at $\infty$ it must be zero).

The image consists of those analytic functions in $C\backslash\gamma$ which tend to $0$ as $z\to\infty$, have limits on both sides of $\gamma$ and the jump between these limits is an entire function (analytic in $C$). (At the end-points, the limits do not exist, and we have logarithmic singularities.)

With some regularity of $\gamma$ and $f$ (non necessary analytic) we have $$F^+(z)=F^*(z)-\frac{1}{2}f(z),\quad F^-(z)=F^*(z)+\frac{1}{2}f(z),$$ where $F^*(z)$ is defined for $z\in\gamma$ using the principal value of the integral in (1). The above formula is obtained by subtracting these two relations. But when $f$ is analytic on $\gamma$, the curve can be slightly deformed (preserving its ends), and you get rid of the regularity condition by approximating it with a smooth (or piecewise-linear) curve with the same ends.

If the curve happens to be closed then $F^+=0$ and we recover the result for the closed curve which you cite.

It is interesting that the curve $\gamma$ plays little role in the description of the image: functions from the image have analytic continuation along any curve starting and ending at $\infty$ and not passing through the endpoints of $\gamma$.

References. This material is standard and is included to every Russian undergraduate Complex Variables text, but for some strange reasons none of them is translated into English. A good reference in English which comes to my mind is a translation from the Japanese:

MR1026013
Kaneko, A. Introduction to hyperfunctions. Kluwer, Dordrecht, Tokyo, 1988.

Such an integral over a simple (non-closed) curve is called the Cauchy type integral.

The curve $\gamma$ is oriented, so for a function $F$ defined in $C\backslash\gamma$ and $z\in\gamma$ different from an endpoint, we can talk of the right limit $F^+(z)$ and left limit $F^-(z)$. Then we have Sokhotski formula $$F^-(z)-F^+(z)=f(z),\quad z\in\Gamma$$ As in addition $F(z)\to0,\; z\to\infty$, this shows that your map is injective. The image consists of those analytic functions in $C\backslash\gamma$ which tend to $0$ as $z\to\infty$, have limits on both sides of $\gamma$ and the jump between these limits is an entire function (analytic in $C$). (At the end-points, the limits do not exist, and we have logarithmic singularities.)

With some regularity of $\gamma$ and $f$ (non necessary analytic) we have $$F^+(z)=F^*(z)-\frac{1}{2}f(z),\quad F^-(z)=F^*(z)+\frac{1}{2}f(z),$$ where $F^*(z)$ is the principal value of your integral. The above formula is obtained by subtracting these two relations. But when $f$ is entire, the curve can be slightly deformed (preserving its ends), and you get rid of the regularity condition by approximating it with a smooth (or piecewise-linear) curve with the same ends.

If the curve happens to be closed then $F^+=0$ and we recover the result for the closed curve which you cite.

It is interesting that the curve $\gamma$ plays little role in the description of the image: functions from the image have analytic continuation along any curve starting and ending at $\infty$ and not passing through the endpoints of $\gamma$.

References. This material is standard and is included to every Russian undergraduate Complex Variables text, but for some strange reasons none of them is translated into English. A good reference in English which comes to my mind is a translation from the Japanese:

MR1026013
Kaneko, A. Introduction to hyperfunctions. Kluwer, Dordrecht, Tokyo, 1988.

Such an integral over a simple (non-closed) curve is called the Cauchy type integral. It is convenient to define $$F(z)=\frac{1}{2\pi i}\int_\gamma \frac{f(\zeta)d\zeta}{\zeta-z},\quad z\not\in\gamma.\tag{1}$$

The curve $\gamma$ is oriented, so for our function $F$ defined in $C\backslash\gamma$ and $z\in\gamma$ different from an endpoint, we can talk of the right limit $F^+(z)$ and left limit $F^-(z)$. Then we have Sokhotski formula $$F^-(z)-F^+(z)=f(z),\quad z\in\Gamma$$ As in addition $F(z)\to0,\; z\to\infty$, this shows that your map is injective. The image consists of those analytic functions in $C\backslash\gamma$ which tend to $0$ as $z\to\infty$, have limits on both sides of $\gamma$ and the jump between these limits is an entire function (analytic in $C$). (At the end-points, the limits do not exist, and we have logarithmic singularities.)

With some regularity of $\gamma$ and $f$ (non necessary analytic) we have $$F^+(z)=F^*(z)-\frac{1}{2}f(z),\quad F^-(z)=F^*(z)+\frac{1}{2}f(z),$$ where $F^*(z)$ is defined for $z\in\gamma$ using the principal value of the integral in (1). The above formula is obtained by subtracting these two relations. But when $f$ is entire, the curve can be slightly deformed (preserving its ends), and you get rid of the regularity condition by approximating it with a smooth (or piecewise-linear) curve with the same ends.

If the curve happens to be closed then $F^+=0$ and we recover the result for the closed curve which you cite.

It is interesting that the curve $\gamma$ plays little role in the description of the image: functions from the image have analytic continuation along any curve starting and ending at $\infty$ and not passing through the endpoints of $\gamma$.

References. This material is standard and is included to every Russian undergraduate Complex Variables text, but for some strange reasons none of them is translated into English. A good reference in English which comes to my mind is a translation from the Japanese:

MR1026013
Kaneko, A. Introduction to hyperfunctions. Kluwer, Dordrecht, Tokyo, 1988.

Such an integral over a simple (non-closed) curve is called the Cauchy type integral.

The curve $\gamma$ is oriented, so for a function $F$ defined in $C\backslash\gamma$ and $z\in\gamma$ different from an endpoint, we can talk of the right limit $F^+(z)$ and left limit $F^-(z)$. Then we have Sokhotski formula $$F^-(z)-F^+(z)=f(z),\quad z\in\Gamma$$ As in addition $F(z)\to0,\; z\to\infty$, this shows that your map is injective. The image consists of those analytic functions in $C\backslash\gamma$ which tend to $0$ as $z\to\infty$, have limits on both sides of $\gamma$ and the jump between these limits is an entire function (analytic in $C$). (At the end-points, the limits do not exist, and we have logarithmic singularities.)

With some regularity of $\gamma$ and $f$ (non necessary analytic) we have $$F^+(z)=F(z)-\frac{1}{2}f(z),\quad F^-(z)=F^*(z)+\frac{1}{2}f(z),$$ where $F^*(z)$ is the principal value of your integral. The above formula is obtained by subtracting these two relations. But when $f$ is entire, the curve can be slightly deformed (preserving its ends), and you get rid of the regularity condition by approximating it with a smooth (or piecewise-linear) curve with the same ends.

If the curve happens to be closed then $F^+=0$ and we recover the result for the closed curve which you cite.

It is interesting that the curve $\gamma$ plays little role in the description of the image: functions from the image have analytic continuation along any curve starting and ending at $\infty$ and not passing through the endpoints of $\gamma$.

References. This material is standard and is included to every Russian undergraduate Complex Variables text, but for some strange reasons none of them is translated into English. A good reference in English which comes to my mind is a translation from the Japanese:

MR1026013
Kaneko, A. Introduction to hyperfunctions. Kluwer, Dordrecht, Tokyo, 1988.

Such an integral over a simple (non-closed) curve is called the Cauchy type integral.

The curve $\gamma$ is oriented, so for a function $F$ defined in $C\backslash\gamma$ and $z\in\gamma$ different from an endpoint, we can talk of the right limit $F^+(z)$ and left limit $F^-(z)$. Then we have Sokhotski formula $$F^-(z)-F^+(z)=f(z),\quad z\in\Gamma$$ As in addition $F(z)\to0,\; z\to\infty$, this shows that your map is injective. The image consists of those analytic functions in $C\backslash\gamma$ which tend to $0$ as $z\to\infty$, have limits on both sides of $\gamma$ and the jump between these limits is an entire function (analytic in $C$). (At the end-points, the limits do not exist, and we have logarithmic singularities.)

With some regularity of $\gamma$ and $f$ (non necessary analytic) we have $$F^+(z)=F^*(z)-\frac{1}{2}f(z),\quad F^-(z)=F^*(z)+\frac{1}{2}f(z),$$ where $F^*(z)$ is the principal value of your integral. The above formula is obtained by subtracting these two relations. But when $f$ is entire, the curve can be slightly deformed (preserving its ends), and you get rid of the regularity condition by approximating it with a smooth (or piecewise-linear) curve with the same ends.

If the curve happens to be closed then $F^+=0$ and we recover the result for the closed curve which you cite.

It is interesting that the curve $\gamma$ plays little role in the description of the image: functions from the image have analytic continuation along any curve starting and ending at $\infty$ and not passing through the endpoints of $\gamma$.

References. This material is standard and is included to every Russian undergraduate Complex Variables text, but for some strange reasons none of them is translated into English. A good reference in English which comes to my mind is a translation from the Japanese:

MR1026013
Kaneko, A. Introduction to hyperfunctions. Kluwer, Dordrecht, Tokyo, 1988.