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John
John
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I have an integral of the form $$ I=\int_0^{\infty}{dx}\ln \bigg(1+\exp(-\frac{f(x)}{a})\bigg) $$ where $a$ is a positive constant and $f(x)$ is a regular and positive function such that $0\leq f(x) \leq b$ for some positive constant $b$ and $I$ is finite;finite (for example: $f(x)=x$); moreover $f(x)=x+O(x^3)$ for $x \rightarrow 0$. I have to expand this integral $I=c_0+c_1a+..+c_na^n+O(a^{n+1})$ for $a \rightarrow 0^+$, with the value of the integral beeing dominated by the behaviour of $f(x)$ near $0$. Can anyone give me some hints or references to compute the first coefficients of this expansion?

I have an integral of the form $$ I=\int_0^{\infty}{dx}\ln \bigg(1+\exp(-\frac{f(x)}{a})\bigg) $$ where $a$ is a positive constant and $f(x)$ is a regular function such that $0\leq f(x) \leq b$ for some positive constant $b$ and $I$ is finite; moreover $f(x)=x+O(x^3)$ for $x \rightarrow 0$. I have to expand this integral $I=c_0+c_1a+..+c_na^n+O(a^{n+1})$ for $a \rightarrow 0^+$, with the value of the integral beeing dominated by the behaviour of $f(x)$ near $0$. Can anyone give me some hints or references to compute the first coefficients of this expansion?

I have an integral of the form $$ I=\int_0^{\infty}{dx}\ln \bigg(1+\exp(-\frac{f(x)}{a})\bigg) $$ where $a$ is a positive constant and $f(x)$ is a regular and positive function such that $I$ is finite (for example: $f(x)=x$); moreover $f(x)=x+O(x^3)$ for $x \rightarrow 0$. I have to expand this integral $I=c_0+c_1a+..+c_na^n+O(a^{n+1})$ for $a \rightarrow 0^+$, with the value of the integral beeing dominated by the behaviour of $f(x)$ near $0$. Can anyone give me some hints or references to compute the first coefficients of this expansion?

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John
John
  • 31
  • 4

I have an integral of the form $$ I=\int_0^{\infty}{dx}\ln \bigg(1+\exp(-\frac{f(x)}{a})\bigg) $$ where $a$ is a positive constant and $f(x)$ is a regular function such that $0\leq f(x) \leq b$ for some positive constant $b$; and $I$ is finite; moreover $f(x)=x+O(x^3)$ for $x \rightarrow 0$. I have to expand this integral $I=c_0+c_1a+..+c_na^n+O(a^{n+1})$ for $a \rightarrow 0^+$, with the value of the integral beeing dominated by the behaviour of $f(x)$ near $0$. Can anyone give me some hints or references to compute the first coefficients of this expansion?

I have an integral of the form $$ I=\int_0^{\infty}{dx}\ln \bigg(1+\exp(-\frac{f(x)}{a})\bigg) $$ where $a$ is a positive constant and $f(x)$ is a regular function such that $0\leq f(x) \leq b$ for some positive constant $b$; moreover $f(x)=x+O(x^3)$ for $x \rightarrow 0$. I have to expand this integral $I=c_0+c_1a+..+c_na^n+O(a^{n+1})$ for $a \rightarrow 0^+$, with the value of the integral beeing dominated by the behaviour of $f(x)$ near $0$. Can anyone give me some hints or references to compute the first coefficients of this expansion?

I have an integral of the form $$ I=\int_0^{\infty}{dx}\ln \bigg(1+\exp(-\frac{f(x)}{a})\bigg) $$ where $a$ is a positive constant and $f(x)$ is a regular function such that $0\leq f(x) \leq b$ for some positive constant $b$ and $I$ is finite; moreover $f(x)=x+O(x^3)$ for $x \rightarrow 0$. I have to expand this integral $I=c_0+c_1a+..+c_na^n+O(a^{n+1})$ for $a \rightarrow 0^+$, with the value of the integral beeing dominated by the behaviour of $f(x)$ near $0$. Can anyone give me some hints or references to compute the first coefficients of this expansion?

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John
John
  • 31
  • 4

Expansion of an integral

I have an integral of the form $$ I=\int_0^{\infty}{dx}\ln \bigg(1+\exp(-\frac{f(x)}{a})\bigg) $$ where $a$ is a positive constant and $f(x)$ is a regular function such that $0\leq f(x) \leq b$ for some positive constant $b$; moreover $f(x)=x+O(x^3)$ for $x \rightarrow 0$. I have to expand this integral $I=c_0+c_1a+..+c_na^n+O(a^{n+1})$ for $a \rightarrow 0^+$, with the value of the integral beeing dominated by the behaviour of $f(x)$ near $0$. Can anyone give me some hints or references to compute the first coefficients of this expansion?