In this chapter we state and solve some Cauchy type problems in the complex domain formulated in terms of associated integro-differential operators \(\mathbb{L}_{{s + 1/2}}^{*}\left( {s = 1,2,...} \right) \) and \( {\mathbb{L}^ * }_{s + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}\) (s= 1, 2,…) of fractional order. We represent explicitly the solutions of these problems by the Mittag-Leffler type function \( {E_{s + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}\) (z;µ), and we prove an analog of the classical Lagrange formula for these solutions. Then we state some special boundary value problems in the complex domain by means of the mentioned operators. Namely, we assume that the solutions of the mentioned Cauchy type problems satisfy some boundary conditions at the ends of the sum of odd segments $$\gamma 2s + \left( \sigma \right) = \mathop{{\mathop{ \cup }\limits_{{h \to - s}} }}\limits^{s} \left\{ {z = r \exp \left( {i\pi \left( {h + 1/2} \right)} \right),0 \leqslant r \leqslant \sigma } \right\} $$ situated in the Riemann surface G∞ of Ln z.KeywordsEntire FunctionVector FunctionComplex DomainRiesz BaseCommon EndpointThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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