numbered terms

What is the fifth term in the following sequence? asubcript n =n+asubscript n-1. if a1 equals -2, for n greater than or equal to 2.

Jordan's Lemma

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Harmonic function

Please see attached. --- Let (attached) be a function that is analytic and not constant throughtout a bounded domain (attached) and continuous (attached) on its boundary (here domain is an open connected set). Prove, by considering (attached) , that the component function (attached) has a minimum value in the compact region (attached) which occurs on (attached) and never in (attached). Use this result to formulate and prove a minimum principle for harmonic functions. -- (See attached file for full problem description)

Integration of uniformly convergent series

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Removable singularity

(See attached file for full problem description) --- Let have an isolated singularity at and suppose that is bounded in some punctured neighborhood of . Prove directly from the integral formula for the Laurent coefficients that for all j = 1,2,3,..., i.e. must have a removable singularity at . The integral formula for the Laurent coefficients (no need to prove): where C is any positively oriented simple contour lying in the annulus and containing in its interior. --- (See attached file for full problem description)