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I find the notation $ i{\infty}$ a lot in complex analysis e.g. $\int_{0}^{i\infty } f(z) dz $ and also when estimating some functions. My question here is :

What is the mathematical meaning of the titled notation ? and Does it indicate large imaginary part with a null real part ?

Note : I hope someone gives me clear explanations to understand it in the view of complex analysis and gives me reasons of using it especially in the calculations of integral and measure theory .

Thank you for any help

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    $\begingroup$ $i\infty$ is used to refer to something growing large along the positive imaginary axis. For example $\int_0^{i\infty} f(z) dz$ is the same as $\int_0^\infty f(it) d(it) = i \int_0^\infty f(it) dt$. $\endgroup$ – Ian Nov 9 '16 at 14:14
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    $\begingroup$ $\lim_{x\to\infty} ix$ for $x\in\mathbb{R}$ $\endgroup$ – zahbaz Nov 10 '16 at 2:12
  • $\begingroup$ en.wikipedia.org/wiki/Methods_of_contour_integration and read a complex analysis course $\endgroup$ – reuns Nov 16 '16 at 19:49
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This is the notation sometimes used to refer to an unbounded pure imaginary number. In the example you cite,

$$\int_0^{i\infty}f(z)\,dz = \lim_{m\to\infty}\int_0^{i m}f(z)\,dz$$

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