# Region of convergence of a Laplace transform

• I
• mjtsquared
In summary, the Laplace transform of a function is defined as the integral of the function multiplied by the exponential of a complex variable. The region of convergence for a Laplace transform is a right half-plane where the real part of the complex variable is greater than a certain value. In some cases, the Laplace transform can be extended to a larger region through analytic continuation. In the case of the Laplace transform of 1, it exists at s=i since it is analytic everywhere in the complex plane except at s=0. This is due to the fact that 1/s has a Taylor series that converges for |s-i|<1. In basic electronics problems, the Laplace transform is often a rational function, making

#### mjtsquared

If a Laplace transform has a region of convergence starting at Re(s)=0, does the Laplace transform evaluated at the imaginary axis exist? I.e. say that the Laplace transform of 1 is 1/s. Does this Laplace transform exist at say s=i?

I know this thread is old, but in case you still interested here is an answer.

I am assuming you are using the single-sided Laplace transform, with the standard definition
$$F(s) = \int_0^\infty f(t) e^{-s t} dt$$
so the region of convergence a right half-plane ##\Re(s)>s_0## (where the notation ##\Re(s)## indicates the real part of ##s##). This means that ##F(s)## is analytic in that half-plane, and in general is not defined outside of that half-plane. However, in many cases ##F(s)## can be extended so that it is analytic in a larger region of the complex plane (this is called analytic continuation). In your example you have ##F(s) = 1/s## with ##\Re(s)>0##. Since ##1/s## is analytic everywhere in the complex plane except at ##s=0##, ##1/s## is (trivially) the analytic continuation of ##F(s)## into the entire plane except the origin. So yes, in your example ##1/s## is defined at ##s=i##.

If the term analytic is new to you, a function ##F(s)## is called analytic at a point if there is a converging Taylor series about that point. In your example, ##1/s## has a Taylor series about ##i##: ##F(s) = -i + (s-i) + i (s-i)^2 + \ldots##, which converges for ##|s-i|<1## .

Note that in some cases the process of analytic continuation is much more complicatd. However, in the types of problems you are likely to find in basic electronics, ##F(s)## is often a rational function, so is trivially its own analytic continuation into the entire complex plane except at the poles of ##F(s)##.

Jason
{"ft_lang_label":"__label__en","ft_lang_prob":0.9127951,"math_prob":0.97180116,"size":2421,"snap":"2023-40-2023-50","text_gpt3_token_len":607,"char_repetition_ratio":0.1481175,"word_repetition_ratio":0.029411765,"special_character_ratio":0.26517966,"punctuation_ratio":0.08179959,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99126285,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-01T10:38:15Z\",\"WARC-Record-ID\":\"<urn:uuid:5db14cbe-a5dc-41df-905b-57b373596ab7>\",\"Content-Length\":\"67711\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e5e8ca4b-97b1-4591-891b-c9db34c5a1b4>\",\"WARC-Concurrent-To\":\"<urn:uuid:26192ae6-85d1-48a0-b00c-2f5261aedc0b>\",\"WARC-IP-Address\":\"104.26.14.132\",\"WARC-Target-URI\":\"https://www.physicsforums.com/threads/region-of-convergence-of-a-laplace-transform.949184/\",\"WARC-Payload-Digest\":\"sha1:ZPNDSE7XEYRLOLWYHI2W7XHUDSQ6IAXN\",\"WARC-Block-Digest\":\"sha1:A3D7U3JA5S7XVYN3QGNPGIMXP6CLUSDY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100286.10_warc_CC-MAIN-20231201084429-20231201114429-00138.warc.gz\"}"}