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There are two aspects of this question which I would like to comment, firstly the question of in which sense the integrals are to be understood (they clearly diverge in any classical sense) and secondly how to compute them. I will begin with the second one.

It seems to me that the simplest approach is to use partial fractions, more precisely, those for $\dfrac {k}{k(k^2+\alpha^2)}$ and $\dfrac {\alpha^2}{k(k^2+\alpha^2)}$ which are very simple to calculate, to reduce to the case of the FT of $\frac 1k$ and $\frac 1{k-a}$, which can be easily found (basically the sign function up to the usual struggle with constants).

However, these remain mere formal manipulations, without a corresponding foundation to provide a basis for the integral and the manipulations used to compute them. The integrals are all divergent partial integrals of „functions“ of two variables and the appropriate framework is that of partial integrals in the distributional sense of distributions of several variables. Fortunately such a theory was developed in the 50‘s and 60‘s of the last century. Further it is completely elementary, at the level of a first year analysis course without functional analysis. For all practical purposes one only needs three facts:

1. If an integral converges in the classical sense, then also in the distributional one;

2. Fubinís theorem on exchanging the order of integration always holds;

3. the freshman‘s dream theorems on exchanging limits or differentiation (Euler) always hold.

One remark on the function $\frac 1k$. This can be regarded as a distribution by simply defining it to be the distributional derivative of the locally integrable function $\ln |k|$.

A lucid treatment of this theory can be found in the treatise „Theory of Distributions“ at the site of the late portuguese mathematician J. Sebastião e Silva (jss100.campus.ciencias.ulisboa.pt). The most relevant are the last two chapters on partial integrals, with applications to the F.T.

There are two aspects of this question which I would like to comment, firstly the question of in which sense the integrals are to be understood (they clearly diverge in any classical sense) and secondly how to compute them. I will begin with the second one.

It seems to me that the simplest approach is to use partial fractions, more precisely, those for $\dfrac {k}{(k^2+\alpha^2)}$ and $\dfrac {\alpha^2}{k(k^2+\alpha^2)}$ which are very simple to calculate, to reduce to the case of the FT of $\frac 1k$ and $\frac 1{k-a}$, which can be easily found (basically the sign function up to the usual struggle with constants).

However, these remain mere formal manipulations, without a corresponding foundation to provide a basis for the integral and the manipulations used to compute them. The integrals are all divergent partial integrals of „functions“ of two variables and the appropriate framework is that of partial integrals in the distributional sense of distributions of several variables. Fortunately such a theory was developed in the 50‘s and 60‘s of the last century. Further it is completely elementary, at the level of a first year analysis course without functional analysis. For all practical purposes one only needs three facts:

1. If an integral converges in the classical sense, then also in the distributional one;

2. Fubini‘s theorem on exchanging the order of integration always holds;

3. the freshman‘s dream theorems on exchanging limits or differentiating under the integral sigm (Euler) always hold.

Precise statements can be found in the reference below.

One remark on the function $\frac 1k$. This can be regarded as a distribution by simply defining it to be the distributional derivative of the locally integrable function $\ln |k|$. There are, of course, other equivalent possibilities, e.g., as a principal value, but this seems to be the simplest one.

A lucid treatment of this theory can be found in the treatise „Theory of Distributions“ at the site of the late portuguese mathematician J. Sebastião e Silva (jss100.campus.ciencias.ulisboa.pt). The most relevant chapters are the last two, on partial integrals, with applications to the F.T.

There are two aspects of this question which I would like to comment, firstly the question of in which sense the integrals are to be understood (they clearly diverge in any classical sense) and secondly how to compute them. I will begin with the second one.

It seems to me that the simplest approach is to use partial fractions, more precisely, those for $\dfrac {k}{k(k^2+\alpha^2)}$ and $\dfrac {\alpha^2}{k(k^2+\alpha^2)}$ which are very simple to calculate, to reduce to the case of the FT of $\frac 1k$ and $\frac 1{k-a}$, which can be easily found (basically the sign function up to the usual struggle with constants).

However, these remain mere formal manipulations, without a corresponding foundation to provide a basis for the integral and the manipulations used to compute them. The integrals are all divergent partial integrals of „functions“ of two variables and the appropriate framework is that of partial integrals in the distributional sense of distributions of several variables. Fortunately such a theory was developed in the 50‘s and 60‘s of the last century. Further it is completely elementary, at the level of a first year analysis course without functional analysis. For all practical purposes one only needs three facts:

1. If an integral converges in the classical sense, then also in the distributional one;

2. Fubinís theorem on exchanging the order of integration always holds;

3. the freshman‘s dream theorems on exchanging limits or differentiation (Euler) always hold.

One remark on the function $1k$. This can be regarded as a distribution by simply defining it to be the distributional derivative of the locally integrable function $\ln |k|$.

A lucid treatment of this theory can be found in the treatise „Theory of Distributions“ at the site of the late portuguese mathematician J. Sebastião e Silva (jss100.campus.ciencias.ulisboa.pt). The most relevant are the last two chapters on partial integrals, with applications to the F.T.

There are two aspects of this question which I would like to comment, firstly the question of in which sense the integrals are to be understood (they clearly diverge in any classical sense) and secondly how to compute them. I will begin with the second one.

It seems to me that the simplest approach is to use partial fractions, more precisely, those for $\dfrac {k}{k(k^2+\alpha^2)}$ and $\dfrac {\alpha^2}{k(k^2+\alpha^2)}$ which are very simple to calculate, to reduce to the case of the FT of $\frac 1k$ and $\frac 1{k-a}$, which can be easily found (basically the sign function up to the usual struggle with constants).

However, these remain mere formal manipulations, without a corresponding foundation to provide a basis for the integral and the manipulations used to compute them. The integrals are all divergent partial integrals of „functions“ of two variables and the appropriate framework is that of partial integrals in the distributional sense of distributions of several variables. Fortunately such a theory was developed in the 50‘s and 60‘s of the last century. Further it is completely elementary, at the level of a first year analysis course without functional analysis. For all practical purposes one only needs three facts:

1. If an integral converges in the classical sense, then also in the distributional one;

2. Fubinís theorem on exchanging the order of integration always holds;

3. the freshman‘s dream theorems on exchanging limits or differentiation (Euler) always hold.

One remark on the function $\frac 1k$. This can be regarded as a distribution by simply defining it to be the distributional derivative of the locally integrable function $\ln |k|$.

A lucid treatment of this theory can be found in the treatise „Theory of Distributions“ at the site of the late portuguese mathematician J. Sebastião e Silva (jss100.campus.ciencias.ulisboa.pt). The most relevant are the last two chapters on partial integrals, with applications to the F.T.

There are two aspects of this question which I would like to comment, firstly the question of in which sense the integrals are to be understood (they clearly diverge in any classical sense) and secondly how to compute them. I will begin with the second one.

It seems to me that the simplest approach is to use partial fractions, more precisely, those for $\dfrac {k}{k(k^2+\alpha^2)}$ and $\dfrac {\alpha^2}{k(k^2+\alpha^2)}$ which are very simple to calculate, to reduce to the case of the FT of $\frac 1k$ and $\frac 1{k-a}$, which can be easily found (basically the sign function up to the usual struggle with constants).

However, these remain mere formal manipulations, without a corresponding foundation to provide a basis for the integral and the manipulations used to compute them. The integrals are all divergent partial integrals of „functions“ of two variables and the appropriate framework is that of partial integrals in the distributional sense of distributions of several variables. Fortunately such a theory was developed in the 50‘s and 60‘s of the last century. Further it is completely elementary, at the level of a first year analysis course without functional analysis.

There are two aspects of this question which I would like to comment, firstly the question of in which sense the integrals are to be understood (they clearly diverge in any classical sense) and secondly how to compute them. I will begin with the second one.

It seems to me that the simplest approach is to use partial fractions, more precisely, those for $\dfrac {k}{k(k^2+\alpha^2)}$ and $\dfrac {\alpha^2}{k(k^2+\alpha^2)}$ which are very simple to calculate, to reduce to the case of the FT of $\frac 1k$ and $\frac 1{k-a}$, which can be easily found (basically the sign function up to the usual struggle with constants).

However, these remain mere formal manipulations, without a corresponding foundation to provide a basis for the integral and the manipulations used to compute them. The integrals are all divergent partial integrals of „functions“ of two variables and the appropriate framework is that of partial integrals in the distributional sense of distributions of several variables. Fortunately such a theory was developed in the 50‘s and 60‘s of the last century. Further it is completely elementary, at the level of a first year analysis course without functional analysis. For all practical purposes one only needs three facts:

1. If an integral converges in the classical sense, then also in the distributional one;

2. Fubinís theorem on exchanging the order of integration always holds;

3. the freshman‘s dream theorems on exchanging limits or differentiation (Euler) always hold.

One remark on the function $1k$. This can be regarded as a distribution by simply defining it to be the distributional derivative of the locally integrable function $\ln |k|$.

A lucid treatment of this theory can be found in the treatise „Theory of Distributions“ at the site of the late portuguese mathematician J. Sebastião e Silva (jss100.campus.ciencias.ulisboa.pt). The most relevant are the last two chapters on partial integrals, with applications to the F.T.